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Common Limits


Reference table of common limit identities. Try puzzle mode to drill, or read the full limits explanation →

Common limits tool

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Basic limits
lim⁡x→ac\displaystyle\lim_{x \to a} c
Constant
↓
cc
Basic limits
lim⁡x→ax\displaystyle\lim_{x \to a} x
Identity
↓
aa
Basic limits
lim⁡x→axn\displaystyle\lim_{x \to a} x^n
Power at a point
↓
ana^n
Trigonometric limits
lim⁡x→0sin⁡(x)x\displaystyle\lim_{x \to 0} \dfrac{\sin(x)}{x}
Sine over x
↓
11
Trigonometric limits
lim⁡x→0tan⁡(x)x\displaystyle\lim_{x \to 0} \dfrac{\tan(x)}{x}
Tangent over x
↓
11
Trigonometric limits
lim⁡x→01−cos⁡(x)x\displaystyle\lim_{x \to 0} \dfrac{1 - \cos(x)}{x}
(1 - cos x) over x
↓
00
Trigonometric limits
lim⁡x→01−cos⁡(x)x2\displaystyle\lim_{x \to 0} \dfrac{1 - \cos(x)}{x^2}
(1 - cos x) over x squared
↓
12\dfrac{1}{2}
Trigonometric limits
lim⁡x→0sin⁡(kx)x\displaystyle\lim_{x \to 0} \dfrac{\sin(kx)}{x}
Sine of kx over x
↓
kk
Trigonometric limits
lim⁡x→∞sin⁡(x)x\displaystyle\lim_{x \to \infty} \dfrac{\sin(x)}{x}
Sine over x at infinity
↓
00
Exponential limits
lim⁡x→∞(1+1x)x\displaystyle\lim_{x \to \infty} \left(1 + \dfrac{1}{x}\right)^x
Definition of e at infinity
↓
ee
Exponential limits
lim⁡x→0(1+x)1/x\displaystyle\lim_{x \to 0} (1 + x)^{1/x}
Definition of e at zero
↓
ee
Exponential limits
lim⁡x→∞(1+ax)x\displaystyle\lim_{x \to \infty} \left(1 + \dfrac{a}{x}\right)^x
Scaled definition of e
↓
eae^a
Exponential limits
lim⁡x→0ex−1x\displaystyle\lim_{x \to 0} \dfrac{e^x - 1}{x}
e to the x minus one over x
↓
11
Exponential limits
lim⁡x→0ax−1x\displaystyle\lim_{x \to 0} \dfrac{a^x - 1}{x}
a to the x minus one over x
↓
ln⁡(a)\ln(a)
Logarithmic limits
lim⁡x→0ln⁡(1+x)x\displaystyle\lim_{x \to 0} \dfrac{\ln(1 + x)}{x}
ln(1+x) over x
↓
11
Logarithmic limits
lim⁡x→∞ln⁡(x)x\displaystyle\lim_{x \to \infty} \dfrac{\ln(x)}{x}
ln x over x at infinity
↓
00
Logarithmic limits
lim⁡x→0+xln⁡(x)\displaystyle\lim_{x \to 0^+} x\ln(x)
x times ln x at 0+
↓
00
Limits at infinity
lim⁡x→∞1x\displaystyle\lim_{x \to \infty} \dfrac{1}{x}
Reciprocal at infinity
↓
00
Limits at infinity
lim⁡x→∞1xn\displaystyle\lim_{x \to \infty} \dfrac{1}{x^n}
Reciprocal power at infinity
↓
00
Limits at infinity
lim⁡x→∞xn\displaystyle\lim_{x \to \infty} x^n
Power at infinity
↓
∞\infty
Limits at infinity
lim⁡x→∞xnex\displaystyle\lim_{x \to \infty} \dfrac{x^n}{e^x}
Power over exponential
↓
00
Limits at infinity
lim⁡x→∞ln⁡(x)xn\displaystyle\lim_{x \to \infty} \dfrac{\ln(x)}{x^n}
Logarithm over power
↓
00
Limits at infinity
lim⁡x→∞e−x\displaystyle\lim_{x \to \infty} e^{-x}
Negative exponential
↓
00
Limits at infinity
lim⁡x→∞x1/x\displaystyle\lim_{x \to \infty} x^{1/x}
x to the one over x
↓
11
One-sided limits
lim⁡x→0+1x\displaystyle\lim_{x \to 0^+} \dfrac{1}{x}
Reciprocal from the right
↓
+∞+\infty
One-sided limits
lim⁡x→0−1x\displaystyle\lim_{x \to 0^-} \dfrac{1}{x}
Reciprocal from the left
↓
−∞-\infty
One-sided limits
lim⁡x→0+x\displaystyle\lim_{x \to 0^+} \sqrt{x}
Square root from the right
↓
00
One-sided limits
lim⁡x→0+xx\displaystyle\lim_{x \to 0^+} x^x
x to the x from the right
↓
11

Families of limits

Click a family to highlight its entries in the table above.

c

Basic limits

Constant, identity, and other elementary limits that follow from continuity.

3 matchesClick to highlight
sin

Trigonometric limits

Classical trigonometric limits — the foundation of every trig derivative.

6 matchesClick to highlight
e^x

Exponential limits

Limits that define or relate to the constant ee.

5 matchesClick to highlight
ln

Logarithmic limits

Limits involving the natural logarithm.

3 matchesClick to highlight
∞

Limits at infinity

End behavior of polynomial, rational, exponential, and logarithmic expressions.

7 matchesClick to highlight
0⁺

One-sided limits

Limits approached from the right or left only — useful where the two-sided limit fails to exist.

4 matchesClick to highlight

Limit laws and theorems

The rules that combine, transform, and evaluate limits.

+

Sum and difference

The limit of a sum (or difference) is the sum (or difference) of the limits, provided both exist.

lim⁡x→a(f(x)±g(x))=lim⁡x→af(x)±lim⁡x→ag(x)\displaystyle\lim_{x \to a}\bigl(f(x) \pm g(x)\bigr) = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)
×

Product and constant multiple

The limit of a product is the product of the limits. Constants pull out of the limit.

lim⁡x→a(f(x) g(x))=lim⁡x→af(x)⋅lim⁡x→ag(x)\displaystyle\lim_{x \to a}\bigl(f(x)\,g(x)\bigr) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)
÷

Quotient

The limit of a quotient is the quotient of the limits, provided the denominator's limit is not zero.

lim⁡x→af(x)g(x)=lim⁡x→af(x)lim⁡x→ag(x),lim⁡g≠0\displaystyle\lim_{x \to a}\dfrac{f(x)}{g(x)} = \dfrac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad \lim g \neq 0
≈

Squeeze theorem

If f(x)≤g(x)≤h(x)f(x) \leq g(x) \leq h(x) near aa and lim⁡f=lim⁡h=L\lim f = \lim h = L, then lim⁡g=L\lim g = L as well. Useful when direct evaluation is hard but bounds are easy.

lim⁡x→0x2sin⁡(1/x)=0since−x2≤x2sin⁡(1/x)≤x2\displaystyle\lim_{x \to 0} x^2 \sin(1/x) = 0 \quad \text{since} \quad -x^2 \leq x^2\sin(1/x) \leq x^2
L

L\'Hôpital\'s rule

For indeterminate forms 0/00/0 or ∞/∞\infty/\infty, differentiating top and bottom separately gives a limit of the same value (when it exists).

lim⁡x→af(x)g(x)=lim⁡x→af(ˊx)g(ˊx)\displaystyle\lim_{x \to a}\dfrac{f(x)}{g(x)} = \lim_{x \to a}\dfrac{f\'(x)}{g\'(x)}
C

Continuity

A function is continuous at aa exactly when the limit equals the value. Every polynomial, rational, exponential, logarithmic, and trig function is continuous on its domain.

f continuous at a  ⟺  lim⁡x→af(x)=f(a)\displaystyle f \text{ continuous at } a \iff \lim_{x \to a} f(x) = f(a)
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