What is end behavior?

End behavior is what a function’s output does at the far left and far right of its graph — as x → −∞ and as x → ∞. Each end can rise (f(x) → ∞), fall (f(x) → −∞), level off at a horizontal asymptote (f(x) → a number), or oscillate with no limit. “End behaviour” is the British spelling of the same idea. Right end behavior means x → ∞; left end behavior means x → −∞.

How do you find the end behavior of a function?

Find the term that grows fastest and ignore the rest. For a polynomial, that is the leading term aₙxⁿ (use the leading coefficient test). For a rational function, divide the leading terms of the numerator and denominator. For mixed functions, use the growth order: exponentials beat powers of x, and powers of x beat logarithms. Then write the result for x → −∞ and x → ∞. The calculator above does all three and checks the answer by evaluating f(x) at large inputs.

What is the leading coefficient test?

It reads a polynomial’s end behavior from its degree n and leading coefficient aₙ. Even degree: both ends point the same way — up if aₙ > 0, down if aₙ 0, up on the left and down on the right if aₙ < 0. The other terms only change the middle of the graph.

How does the degree of a polynomial affect its end behavior?

The degree decides whether the ends match. An even power such as x² or x⁴ is positive for both large positive and large negative x, so both ends go the same direction. An odd power such as x³ keeps the sign of x, so the ends go opposite directions. The leading coefficient’s sign then flips the picture. A higher degree makes the ends steeper but does not change their direction.

What is the end behavior of an even-degree polynomial with a negative leading coefficient?

Both ends fall: as x → −∞, f(x) → −∞ and as x → ∞, f(x) → −∞, like an upside-down parabola. Examples are −x², −x⁴ + 4x² + x − 5, −7x⁶ − 4x − 8 and this page’s default example −2x⁴ + 3x² − 1.

What is the end behavior of an odd-degree polynomial with a positive leading coefficient?

It falls on the left and rises on the right: as x → −∞, f(x) → −∞ and as x → ∞, f(x) → ∞, like y = x³. Examples are x³ − 4x, (x − 2)²(x + 3) and 3x⁵ − 2x² + x. With a negative leading coefficient (−2x³ + 5x² − 1, −4x⁵ + 3x³ − 2x + 7, −5x⁷ + 6x²) it is the reverse: up on the left, down on the right.

How do you write end behavior in limit notation?

Write one limit for each end: lim(x→−∞) f(x) = … and lim(x→∞) f(x) = …. For f(x) = 3x⁵ − 2x² + x the leading term 3x⁵ is odd with a positive coefficient, so lim(x→−∞) f(x) = −∞ and lim(x→∞) f(x) = ∞. The arrow form says the same thing: as x → −∞, f(x) → −∞; as x → ∞, f(x) → ∞. If an end levels off, the limit is the asymptote’s value, e.g. lim(x→∞) 1/x = 0.

What is the end behavior of f(x) = x⁴ − 25x²?

Both ends rise: lim(x→±∞) f(x) = ∞, because the leading term x⁴ has even degree and a positive coefficient. The graph crosses the x-axis at −5 and 5 and touches it at 0 (a double zero), so it looks like a “W”. The same holds for x⁴ − 16x², with zeros at −4, 0 (double) and 4.

What is the end behavior of f(x) = (x − 2)²(x + 3)?

Expand only the leading terms: x² · x = x³, so the degree is 3 and the leading coefficient is 1. The graph falls to the left and rises to the right (f(x) → −∞ as x → −∞, f(x) → ∞ as x → ∞). It crosses the axis at x = −3 and bounces off it at x = 2, a zero of multiplicity 2.

What is the end behavior of f(x) = −6x²(5x + 3)?

The leading term is −6x² · 5x = −30x³: odd degree, negative coefficient. So the graph rises to the left and falls to the right: as x → −∞, f(x) → ∞; as x → ∞, f(x) → −∞. Its zeros are x = 0 (double, the graph touches) and x = −3/5.

Do x² − 2x + 2 and −x² − 3x + 3 have the same end behavior?

No. Both have degree 2 (even), but x² − 2x + 2 has a positive leading coefficient, so both ends rise, while −x² − 3x + 3 has a negative leading coefficient, so both ends fall. Two polynomials share end behavior only when their degrees have the same parity (both even or both odd) and their leading coefficients have the same sign.

What is the end behavior of x⁶ − 3x⁵ − 17x³, −7x⁶ − 4x − 8 and −4x⁵ + 3x³ − 2x + 7?

x⁶ − 3x⁵ − 17x³: even degree, positive coefficient, both ends rise. −7x⁶ − 4x − 8: even degree, negative coefficient, both ends fall. −4x⁵ + 3x³ − 2x + 7: odd degree, negative coefficient, up on the left and down on the right. g(x) = −5x⁷ + 6x² behaves the same way as the last one.

How do you find the end behavior of a rational function?

Compare the degree n of the numerator with the degree m of the denominator. If n m, there is no horizontal asymptote: the ends follow (aₙ/bₘ)xⁿ⁻ᵐ, and when n = m + 1 long division gives a slant asymptote. Vertical asymptotes and holes affect the middle of the graph, not the ends.

What is the end behavior of f(x) = (x² + 4)/(x − 1)?

The numerator’s degree is one higher, so the ends behave like x²/x = x: the graph falls to the left and rises to the right. Long division gives f(x) = x + 1 + 5/(x − 1), so it hugs the slant asymptote y = x + 1, with a vertical asymptote at x = 1. If you type it as x2+4/x-1 without parentheses, it means x² + 4/x − 1, a different function.

What is the end behavior of (2x + 1)/(3x − 4) and (3x − 5)/(x + 1)?

Both have equal degrees, so each end levels off at the ratio of the leading coefficients. (2x + 1)/(3x − 4) → 2/3 as x → ±∞ (from below on the left, from above on the right), with a vertical asymptote at x = 4/3. (3x − 5)/(x + 1) → 3 (from above on the left, from below on the right), with a vertical asymptote at x = −1.

What is the end behavior of (3x² + 4)/(x² + 1) and (4x³ + 2x + 5)/x³?

Equal degrees again: (3x² + 4)/(x² + 1) → 3 at both ends and approaches from above, because it equals 3 + 1/(x² + 1). (4x³ + 2x + 5)/x³ = 4 + 2/x² + 5/x³ → 4 at both ends, also from above, with a vertical asymptote at x = 0.

What is the end behavior of f(x) = 1/x?

Both ends flatten onto the x-axis: lim(x→−∞) 1/x = 0 (from below, the values are negative) and lim(x→∞) 1/x = 0 (from above). The line y = 0 is the horizontal asymptote and x = 0 is a vertical asymptote. Every rational function whose numerator degree is lower than its denominator degree does the same.

What is the end behavior of an exponential function?

For f(x) = a·bˣ + k with b > 1 and a > 0, the right end rises to ∞ and the left end levels off at the horizontal asymptote y = k. If 0 < b < 1 (or the exponent is −x) it is mirrored. A negative a flips the rising end downward. Example: f(x) = 2^(x − 3) + 1 → 1 as x → −∞ and → ∞ as x → ∞; at x = 20 it is already 131073.

What is the end behavior of (4eˣ + 2)/(5e²ˣ + 10)?

As x → −∞ every eˣ term shrinks to 0, so f(x) → 2/10 = 1/5. As x → ∞ the denominator’s e²ˣ outgrows the numerator’s eˣ, so f(x) → 0 from above. The graph has two different horizontal asymptotes: y = 1/5 on the left and y = 0 on the right.

What is the end behavior of a logarithmic function?

y = ln(x) or log(x) is only defined for x > 0, so it has no left end at −∞. As x → ∞ it rises to ∞ — slowly, slower than any positive power of x — and as x → 0⁺ it falls to −∞ along the vertical asymptote x = 0. A negative coefficient, as in −ln(x), makes the right end fall instead.

What is an end behavior model, and what is the left-end model of x² − eˣ?

An end behavior model is a simpler function that matches f(x) for large |x|: g is a right-end model if f(x)/g(x) → 1 as x → ∞, and a left-end model if it does so as x → −∞. For x² − eˣ, eˣ dies out on the left, so the left-end model is x² (f → ∞); on the right eˣ dominates, so the right-end model is −eˣ (f → −∞).

How do you find the degree, leading coefficient, end behavior, x- and y-intercepts of a polynomial?

Write it in standard form. The degree is the highest power and the leading coefficient is its coefficient. The end behavior follows from the leading coefficient test, the x-intercepts are the real zeros, and the y-intercept is f(0). For −2x⁴ + 3x² − 1: degree 4, leading coefficient −2, both ends fall, x-intercepts −1, ≈ −0.7071, ≈ 0.7071, 1 (±1 and ±1/√2), y-intercept (0, −1), and at most 3 turning points.

How do you read end behavior from a graph?

Look at the arrows at the far left and far right of the graph. An arrow pointing up means f(x) → ∞, down means f(x) → −∞, and a graph flattening toward a horizontal line y = L means f(x) → L. Describe the left end as x → −∞ and the right end as x → ∞. Wiggles, zeros and turning points in the middle don’t matter.

Can a graph cross its horizontal asymptote?

Yes. A horizontal asymptote only describes the ends, so a graph can cross it in the middle. (x − 1)/(x² + 1) crosses y = 0 at x = 1, and sin(x)/x crosses y = 0 infinitely many times while still approaching 0 at both ends. A graph can never cross a vertical asymptote, because the function is undefined there.

Does every function have end behavior?

Every function defined for large x has some behavior there, but not always a limit. sin(x) and cos(x) keep oscillating between −1 and 1, so their limits at ±∞ do not exist; x·sin(x) oscillates with growing swings. Functions such as ln(x) and √x aren’t defined for large negative x, so they have no left end. The calculator reports each of these cases separately.

How does this end behavior calculator work?

It parses your expression, then expands polynomials exactly and finds the leading term; for rational functions it divides polynomials to get the horizontal, slant or curvilinear asymptote and the side the graph approaches from; for other functions it compares exponential, power and logarithmic growth symbolically. Every answer is checked numerically at large |x|, and the graph, zeros, vertical asymptotes and holes are computed from the same expression.

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Q: How does the degree of a polynomial affect its end behavior?

The degree decides whether the ends match. An even power such as x² or x⁴ is positive for both large positive and large negative x, so both ends go the same direction. An odd power such as x³ keeps the sign of x, so the ends go opposite directions. The leading coefficient’s sign then flips the picture. A higher degree makes the ends steeper but does not change their direction.

Q: What is the end behavior of an even-degree polynomial with a negative leading coefficient?

Both ends fall: as x → −∞, f(x) → −∞ and as x → ∞, f(x) → −∞, like an upside-down parabola. Examples are −x², −x⁴ + 4x² + x − 5, −7x⁶ − 4x − 8 and this page’s default example −2x⁴ + 3x² − 1.

Q: What is the end behavior of an odd-degree polynomial with a positive leading coefficient?

It falls on the left and rises on the right: as x → −∞, f(x) → −∞ and as x → ∞, f(x) → ∞, like y = x³. Examples are x³ − 4x, (x − 2)²(x + 3) and 3x⁵ − 2x² + x. With a negative leading coefficient (−2x³ + 5x² − 1, −4x⁵ + 3x³ − 2x + 7, −5x⁷ + 6x²) it is the reverse: up on the left, down on the right.

Q: How do you write end behavior in limit notation?

Write one limit for each end: lim(x→−∞) f(x) = … and lim(x→∞) f(x) = …. For f(x) = 3x⁵ − 2x² + x the leading term 3x⁵ is odd with a positive coefficient, so lim(x→−∞) f(x) = −∞ and lim(x→∞) f(x) = ∞. The arrow form says the same thing: as x → −∞, f(x) → −∞; as x → ∞, f(x) → ∞. If an end levels off, the limit is the asymptote’s value, e.g. lim(x→∞) 1/x = 0.

Q: What is the end behavior of f(x) = x⁴ − 25x²?

Both ends rise: lim(x→±∞) f(x) = ∞, because the leading term x⁴ has even degree and a positive coefficient. The graph crosses the x-axis at −5 and 5 and touches it at 0 (a double zero), so it looks like a “W”. The same holds for x⁴ − 16x², with zeros at −4, 0 (double) and 4.

Q: What is the end behavior of f(x) = (x − 2)²(x + 3)?

Expand only the leading terms: x² · x = x³, so the degree is 3 and the leading coefficient is 1. The graph falls to the left and rises to the right (f(x) → −∞ as x → −∞, f(x) → ∞ as x → ∞). It crosses the axis at x = −3 and bounces off it at x = 2, a zero of multiplicity 2.

Q: What is the end behavior of f(x) = −6x²(5x + 3)?

The leading term is −6x² · 5x = −30x³: odd degree, negative coefficient. So the graph rises to the left and falls to the right: as x → −∞, f(x) → ∞; as x → ∞, f(x) → −∞. Its zeros are x = 0 (double, the graph touches) and x = −3/5.

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Algebra 2 · Precalculus · Calculus · updated October 2, 2026

End Behavior Calculator: Find the End Behavior of Any Function

Quick answer

End behavior is what f(x) does as x → −∞ (left end) and x → ∞ (right end). For a polynomial, only the leading term aₙxⁿ matters: even degree means both ends go the same way (up if aₙ > 0, down if aₙ < 0); odd degree means opposite ends (down-left, up-right if aₙ > 0). For a rational function, compare degrees: a smaller numerator gives y = 0, equal degrees give y = aₙ/bₘ, and a larger numerator sends the ends to ±∞. Example: f(x) = −2x⁴ + 3x² − 1 has degree 4 and a negative leading coefficient, so lim(x→−∞) f(x) = −∞ and lim(x→∞) f(x) = −∞.

Type any polynomial (expanded or factored), rational, exponential, logarithmic, root or trig function. You get both ends in words, arrows and limit notation, a graph, the leading term, asymptotes, holes, zeros and step-by-step reasoning. Or use the leading coefficient test and the rational degree rules directly.

Polynomials, rationals, exponentials and logs
Limit notation, arrows and plain words
Graph with horizontal, slant and vertical asymptotes
Zeros, holes, y-intercept and steps

Last updated October 2, 2026. Free, no sign-up. Guides: the leading coefficient test · end behavior of a rational function

End Behavior Calculator

Type any function to get both ends in words, arrows and limit notation, with a graph and steps — or use the leading coefficient test and degree rules.

Function f(x)

Use ^ for powers and parentheses for fractions: (x^2+4)/(x-1). Supports e^x, 2^x, ln, log, sqrt, cbrt, |x|, sin, cos, tan, arctan and factored forms like (x-2)^2(x+3).

Reading: f(x) = −2x⁴ + 3x² − 1

Left end ↙ as x → −∞

f(x) → −∞

behaves like −2x⁴

Right end ↘ as x → ∞

f(x) → −∞

behaves like −2x⁴

Both ends fall (down on the left, down on the right). As x → −∞, f(x) → −∞; as x → ∞, f(x) → −∞.

lim(x→−∞) f(x) = −∞

lim(x→∞) f(x) = −∞

Blue: the graph. Green dashed: horizontal or slant asymptote. Red dashed: vertical asymptote. Values far outside the window are clipped.

Degree · leading coefficient

4 · −2

Leading term −2x⁴

Real zeros (x-intercepts)

−1, ≈ −0.7071, ≈ 0.7071, 1

y-intercept

(0, −1)

Turning points

at most 3

degree − 1

Step by step

Leading term: −2x⁴ → degree 4 (even), leading coefficient −2 (negative).
Even degree: both ends point the same way — down, because the leading coefficient is negative.
Check with large x: f(100) ≈ −199970001, f(−100) ≈ −199970001.
Real zeros: x = −1, x = ≈ −0.7071, x = ≈ 0.7071, x = 1; y-intercept (0, −1).

How to find end behavior

End behavior describes what the output f(x) does as the input x heads toward negative infinity (the left end of the graph) and positive infinity (the right end). Each end either rises without bound (f(x) → ∞), falls without bound (f(x) → −∞), levels off toward a horizontal asymptote (f(x) → L), or has no limit because it keeps oscillating. For polynomials the leading term alone decides it; for rational functions you compare the degrees of the numerator and denominator; for exponential and logarithmic functions you compare growth rates.

Polynomial: f(x) ≈ aₙxⁿ as x → ±∞
Rational: f(x) ≈ (aₙxⁿ) ÷ (bₘxᵐ) = (aₙ/bₘ)·xⁿ⁻ᵐ
Growth order: ln x ≪ xᵖ ≪ bˣ (b > 1) as x → ∞

Find the leading term. For −2x⁴ + 3x² − 1 it is −2x⁴: degree 4 (even), coefficient −2 (negative).
Apply the rule. Even and negative means both ends fall: As x → −∞, f(x) → −∞; as x → ∞, f(x) → −∞.
Rational functions: divide the leading terms. (x² + 4)/(x − 1) ≈ x²/x = x, so the ends go −∞ and ∞, and long division gives the slant asymptote y = x + 1.
Exponentials and logs: the fastest-growing piece wins at each end. In 2^(x − 3) + 1 the exponential dies out on the left, leaving lim(x→−∞) f(x) = 1, and takes over on the right: lim(x→∞) f(x) = ∞.
Check: plug in a big number. f(100) for −2x⁴ + 3x² − 1 is about −2 × 10⁸ — negative and huge, confirming the right end falls.

Why it works: for large |x|, the highest power outgrows every other term, so dividing f(x) by the leading term gives a ratio that approaches 1. Zeros, turning points, holes and vertical asymptotes all live in the middle of the graph; they never change the ends.

End behavior chart: the leading coefficient test

Four cases cover every polynomial of degree 1 or more. The arrows show the left end, then the right end.

Degree	Leading coefficient	Arrows	End behavior	Limit notation	Examples
Even	Positive	↖ ↗	Both ends rise (up on the left, up on the right).	lim(x→−∞) f(x) = ∞; lim(x→∞) f(x) = ∞	x², 3x⁴ − x
Even	Negative	↙ ↘	Both ends fall (down on the left, down on the right).	lim(x→−∞) f(x) = −∞; lim(x→∞) f(x) = −∞	−x², −2x⁴ + 3x² − 1
Odd	Positive	↙ ↗	Falls to the left and rises to the right.	lim(x→−∞) f(x) = −∞; lim(x→∞) f(x) = ∞	x³, 3x⁵ − 2x² + x
Odd	Negative	↖ ↘	Rises to the left and falls to the right.	lim(x→−∞) f(x) = ∞; lim(x→∞) f(x) = −∞	−x³, −2x³ + 5x² − 1

Polynomial end behavior examples

Worked answers for the polynomials students ask about most, computed by the calculator above. Factored forms are expanded only far enough to find the leading term.

f(x)	Degree · LC	Ends	As x → −∞	As x → ∞	Real zeros
x⁴ − 25x²	4 · 1	↖ ↗	f(x) → ∞	f(x) → ∞	−5, 0 (×2), 5
(x − 2)²(x + 3)	3 · 1	↙ ↗	f(x) → −∞	f(x) → ∞	−3, 2 (×2)
−6x²(5x + 3)	3 · −30	↖ ↘	f(x) → ∞	f(x) → −∞	−3/5, 0 (×2)
3x⁵ − 2x² + x	5 · 3	↙ ↗	f(x) → −∞	f(x) → ∞	0
−x⁴ + 4x² + x − 5	4 · −1	↙ ↘	f(x) → −∞	f(x) → −∞	≈ 1.2321, ≈ 1.6808
−2x³ + 5x² − 1	3 · −2	↖ ↘	f(x) → ∞	f(x) → −∞	≈ −0.4142, 1/2, ≈ 2.4142
x² − 2x + 2	2 · 1	↖ ↗	f(x) → ∞	f(x) → ∞	none real
−x² − 3x + 3	2 · −1	↙ ↘	f(x) → −∞	f(x) → −∞	≈ −3.7913, ≈ 0.7913
x⁶ − 3x⁵ − 17x³	6 · 1	↖ ↗	f(x) → ∞	f(x) → ∞	0 (×3), ≈ 4.041
−7x⁶ − 4x − 8	6 · −7	↙ ↘	f(x) → −∞	f(x) → −∞	none real
−4x⁵ + 3x³ − 2x + 7	5 · −4	↖ ↘	f(x) → ∞	f(x) → −∞	≈ 1.1944
−5x⁷ + 6x²	7 · −5	↖ ↘	f(x) → ∞	f(x) → −∞	0 (×2), ≈ 1.0371
x⁴ − 16x²	4 · 1	↖ ↗	f(x) → ∞	f(x) → ∞	−4, 0 (×2), 4

End behavior of rational functions: the degree rules

n is the numerator degree, m the denominator degree, aₙ and bₘ their leading coefficients.

Case	Example	Asymptote	Model	Ends
n < m	(x + 1)/(x² − 4)	Horizontal asymptote y = 0	1/x	→ → Levels off toward the horizontal asymptote y = 0 at both ends.
n = m	(3x² + 4)/(x² + 1)	Horizontal asymptote y = 3	3	→ → Levels off toward the horizontal asymptote y = 3 at both ends.
n = m + 1	(x² + 4)/(x − 1)	No horizontal asymptote; slant (oblique) asymptote y = mx + b from long division	x	↙ ↗ Falls to the left and rises to the right.
n ≥ m + 2	−2x³/(x − 5)	No horizontal or slant asymptote; the graph follows a degree-2 polynomial	−2x²	↙ ↘ Both ends fall (down on the left, down on the right).

Why the side of approach matters

After long division, f(x) = quotient + remainder ÷ denominator. The remainder term’s sign tells you whether the graph sits above or below the asymptote at each end — useful for sketching. For (2x + 1)/(3x − 4) = 2/3 + (11/3)/(3x − 4), the extra piece is negative for very negative x and positive for large x, so the graph approaches y = 2/3 from below on the left and from above on the right.

Holes happen when a factor cancels: (x² − 1)/(x − 1) = x + 1 with a hole at x = 1. The hole changes nothing at the ends — the function behaves like the line y = x + 1.

Rational function examples

f(x)	As x → −∞	As x → ∞	End asymptote	Vertical asymptotes	Holes
1/x	f(x) → 0 (from below)	f(x) → 0 (from above)	y = 0	x = 0	—
(2x + 1)/(3x − 4)	f(x) → 2/3 (from below)	f(x) → 2/3 (from above)	y = 2/3	x = 4/3	—
(3x − 5)/(x + 1)	f(x) → 3 (from above)	f(x) → 3 (from below)	y = 3	x = −1	—
(3x² + 4)/(x² + 1)	f(x) → 3 (from above)	f(x) → 3 (from above)	y = 3	—	—
(4x³ + 2x + 5)/x³	f(x) → 4 (from above)	f(x) → 4 (from above)	y = 4	x = 0	—
(x + 1)/(x² − 4)	f(x) → 0 (from below)	f(x) → 0 (from above)	y = 0	x = −2, x = 2	—
(x² + 4)/(x − 1)	f(x) → −∞	f(x) → ∞	y = x + 1	x = 1	—
12x²/(x − 2)	f(x) → −∞	f(x) → ∞	y = 12x + 24	x = 2	—
(x² − 1)/(x − 1)	f(x) → −∞	f(x) → ∞	—	—	x = 1
(x³ + 1)/(x − 2)	f(x) → ∞	f(x) → ∞	y = x² + 2x + 4	x = 2	—

End behavior of exponential, logarithmic and other functions

Exponentials with base greater than 1 rise on the right and die out on the left, leaving any added constant as a horizontal asymptote. Logarithms exist only for x > 0 and rise slowly forever. When terms compete, exponentials beat powers, and powers beat logs.

f(x)	Ends	As x → −∞	As x → ∞	In words
2ˣ	→ ↗	f(x) → 0 (from above)	f(x) → ∞	The graph levels off toward y = 0 on the left and rises to the right.
0.5ˣ	↖ →	f(x) → ∞	f(x) → 0 (from above)	The graph rises to the left and levels off toward y = 0 on the right.
eˣ − 5	→ ↗	f(x) → −5	f(x) → ∞	The graph levels off toward y = −5 on the left and rises to the right.
−3eˣ + 2	→ ↘	f(x) → 2	f(x) → −∞	The graph levels off toward y = 2 on the left and falls to the right.
2^(x − 3) + 1	→ ↗	f(x) → 1	f(x) → ∞	The graph levels off toward y = 1 on the left and rises to the right.
e^(−x)	↖ →	f(x) → ∞	f(x) → 0 (from above)	The graph rises to the left and levels off toward y = 0 on the right.
x·eˣ	→ ↗	f(x) → 0 (from below)	f(x) → ∞	The graph levels off toward y = 0 on the left and rises to the right.
x²·e^(−x)	↖ →	f(x) → ∞	f(x) → 0 (from above)	The graph rises to the left and levels off toward y = 0 on the right.
(4eˣ + 2)/(5e^(2x) + 10)	→ →	f(x) → 1/5	f(x) → 0 (from above)	The graph levels off toward y = 1/5 on the left and levels off toward y = 0 on the right.
x² − eˣ	↖ ↘	f(x) → ∞	f(x) → −∞	Rises to the left and falls to the right.
ln(x)	· ↗	not defined	f(x) → ∞	The graph is not defined far to the left and rises to the right.
log(x) + 2	· ↗	not defined	f(x) → ∞	The graph is not defined far to the left and rises to the right.
ln(x)/x	· →	not defined	f(x) → 0 (from above)	The graph is not defined far to the left and levels off toward y = 0 on the right.
arctan(x)	→ →	f(x) → −π/2 (from above)	f(x) → π/2 (from below)	The graph levels off toward y = −π/2 on the left and levels off toward y = π/2 on the right.

End behavior of parent functions

Function	Ends	Left end	Right end
x	↙ ↗	lim(x→−∞) f(x) = −∞	lim(x→∞) f(x) = ∞
x²	↖ ↗	lim(x→−∞) f(x) = ∞	lim(x→∞) f(x) = ∞
x³	↙ ↗	lim(x→−∞) f(x) = −∞	lim(x→∞) f(x) = ∞
1/x	→ →	lim(x→−∞) f(x) = 0	lim(x→∞) f(x) = 0
1/x²	→ →	lim(x→−∞) f(x) = 0	lim(x→∞) f(x) = 0
√(x)	· ↗	lim(x→−∞) f(x): not defined	lim(x→∞) f(x) = ∞
∛(x)	↙ ↗	lim(x→−∞) f(x) = −∞	lim(x→∞) f(x) = ∞
\|x\|	↖ ↗	lim(x→−∞) f(x) = ∞	lim(x→∞) f(x) = ∞
eˣ	→ ↗	lim(x→−∞) f(x) = 0	lim(x→∞) f(x) = ∞
e^(−x)	↖ →	lim(x→−∞) f(x) = ∞	lim(x→∞) f(x) = 0
ln(x)	· ↗	lim(x→−∞) f(x): not defined	lim(x→∞) f(x) = ∞
sin(x)	∿ ∿	lim(x→−∞) f(x): no limit (oscillates)	lim(x→∞) f(x): no limit (oscillates)
sin(x)/x	→ →	lim(x→−∞) f(x) = 0	lim(x→∞) f(x) = 0
x·sin(x)	∿ ∿	lim(x→−∞) f(x): no limit (oscillates, unbounded)	lim(x→∞) f(x): no limit (oscillates, unbounded)

Even vs odd degree

Even powers turn negatives positive, so both ends of x², x⁴ and x⁶ point the same way. Odd powers keep the sign, so x³ and x⁵ point opposite ways.

Only leading terms matter

For huge x, 3x² + 4 is basically 3x², and x² + 1 is basically x². That is why the ratio of leading coefficients is the horizontal asymptote when degrees match.

Limits say it precisely

lim(x→∞) f(x) = ∞ means f grows without bound; lim(x→∞) f(x) = 3 means it settles at 3. “Does not exist” covers oscillation, like sin x.

Common mistakes

Using the first term as written instead of the highest power: in 5 + 3x − 2x³ the leading term is −2x³.
Forgetting to multiply leading terms in factored form: −6x²(5x + 3) leads with −30x³, not −6x².
Typing a fraction without parentheses: x2+4/x-1 means x² + 4/x − 1; write (x^2+4)/(x-1) for the rational function.
Thinking a horizontal asymptote can’t be crossed — it can, in the middle of the graph.
Giving a left end for ln(x) or √x: they aren’t defined for large negative x.

Sources and how this calculator was checked

The rules follow the OpenStax Precalculus, College Algebra and Calculus textbooks and Paul’s Online Math Notes; definitions match Wolfram MathWorld, and the topic is part of Common Core standard HSF-IF.C.7. Polynomials are expanded exactly and their zeros found numerically; rational functions use exact polynomial long division; other functions use a symbolic growth-rate comparison. Every result is cross-checked by evaluating the function at large positive and negative x, and every table on this page comes from the same code as the calculator.

OpenStax Precalculus 2e — 3.3 Power Functions and Polynomial FunctionsEnd behavior of power functions; the leading term determines a polynomial’s end behavior; at most n − 1 turning points.
OpenStax Precalculus 2e — 3.7 Rational FunctionsHorizontal asymptotes by comparing degrees; slant asymptotes when the numerator degree is one higher; holes and vertical asymptotes.
OpenStax College Algebra 2e — 5.3 Graphs of Polynomial FunctionsZeros and multiplicity (cross vs touch), turning points and end behavior for sketching.
OpenStax Calculus Volume 1 — 4.6 Limits at Infinity and AsymptotesLimit notation for end behavior, end behavior of polynomial, rational, exponential and logarithmic functions; graphs crossing horizontal asymptotes.
Paul’s Online Math Notes — Graphing PolynomialsLeading coefficient test and the four end behavior cases.
Paul’s Online Math Notes — Limits at Infinity, Part ILimits of polynomials and rational functions as x → ±∞ by factoring out the highest power.
Paul’s Online Math Notes — Limits at Infinity, Part IILimits at infinity of exponentials and logarithms, including e^x and ln x.
Khan Academy — End behavior of polynomialsVideo lesson on reading end behavior from the leading term.
Wolfram MathWorld — AsymptoteDefinitions of horizontal, vertical and oblique asymptotes.
Wolfram MathWorld — Leading CoefficientDefinition of the leading coefficient of a polynomial.
Common Core State Standards — HSF-IF.C.77c: graph polynomials showing end behavior; 7d: rational functions with asymptotes and end behavior; 7e: exponential and log functions.

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