Common Limits

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

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Name	Limit expression		Value	Family
Constant	$\displaystyle\lim_{x \to a} c$	=	$c$	Basic limits
Identity	$\displaystyle\lim_{x \to a} x$	=	$a$	Basic limits
Power at a point	$\displaystyle\lim_{x \to a} x^n$	=	$a^n$	Basic limits
Sine over x	$\displaystyle\lim_{x \to 0} \dfrac{\sin(x)}{x}$	=	$1$	Trigonometric limits
Tangent over x	$\displaystyle\lim_{x \to 0} \dfrac{\tan(x)}{x}$	=	$1$	Trigonometric limits
(1 - cos x) over x	$\displaystyle\lim_{x \to 0} \dfrac{1 - \cos(x)}{x}$	=	$0$	Trigonometric limits
(1 - cos x) over x squared	$\displaystyle\lim_{x \to 0} \dfrac{1 - \cos(x)}{x^2}$	=	$\dfrac{1}{2}$	Trigonometric limits
Sine of kx over x	$\displaystyle\lim_{x \to 0} \dfrac{\sin(kx)}{x}$	=	$k$	Trigonometric limits
Sine over x at infinity	$\displaystyle\lim_{x \to \infty} \dfrac{\sin(x)}{x}$	=	$0$	Trigonometric limits
Definition of e at infinity	$\displaystyle\lim_{x \to \infty} \left(1 + \dfrac{1}{x}\right)^x$	=	$e$	Exponential limits
Definition of e at zero	$\displaystyle\lim_{x \to 0} (1 + x)^{1/x}$	=	$e$	Exponential limits
Scaled definition of e	$\displaystyle\lim_{x \to \infty} \left(1 + \dfrac{a}{x}\right)^x$	=	$e^a$	Exponential limits
e to the x minus one over x	$\displaystyle\lim_{x \to 0} \dfrac{e^x - 1}{x}$	=	$1$	Exponential limits
a to the x minus one over x	$\displaystyle\lim_{x \to 0} \dfrac{a^x - 1}{x}$	=	$\ln(a)$	Exponential limits
ln(1+x) over x	$\displaystyle\lim_{x \to 0} \dfrac{\ln(1 + x)}{x}$	=	$1$	Logarithmic limits
ln x over x at infinity	$\displaystyle\lim_{x \to \infty} \dfrac{\ln(x)}{x}$	=	$0$	Logarithmic limits
x times ln x at 0+	$\displaystyle\lim_{x \to 0^+} x\ln(x)$	=	$0$	Logarithmic limits
Reciprocal at infinity	$\displaystyle\lim_{x \to \infty} \dfrac{1}{x}$	=	$0$	Limits at infinity
Reciprocal power at infinity	$\displaystyle\lim_{x \to \infty} \dfrac{1}{x^n}$	=	$0$	Limits at infinity
Power at infinity	$\displaystyle\lim_{x \to \infty} x^n$	=	$\infty$	Limits at infinity
Power over exponential	$\displaystyle\lim_{x \to \infty} \dfrac{x^n}{e^x}$	=	$0$	Limits at infinity
Logarithm over power	$\displaystyle\lim_{x \to \infty} \dfrac{\ln(x)}{x^n}$	=	$0$	Limits at infinity
Negative exponential	$\displaystyle\lim_{x \to \infty} e^{-x}$	=	$0$	Limits at infinity
x to the one over x	$\displaystyle\lim_{x \to \infty} x^{1/x}$	=	$1$	Limits at infinity
Reciprocal from the right	$\displaystyle\lim_{x \to 0^+} \dfrac{1}{x}$	=	$+\infty$	One-sided limits
Reciprocal from the left	$\displaystyle\lim_{x \to 0^-} \dfrac{1}{x}$	=	$-\infty$	One-sided limits
Square root from the right	$\displaystyle\lim_{x \to 0^+} \sqrt{x}$	=	$0$	One-sided limits
x to the x from the right	$\displaystyle\lim_{x \to 0^+} x^x$	=	$1$	One-sided limits

Families of limits

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

Basic limits

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

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sin

Trigonometric limits

Classical trigonometric limits — the foundation of every trig derivative.

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e^x

Exponential limits

Limits that define or relate to the constant $e$ .

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Logarithmic limits

Limits involving the natural logarithm.

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∞

Limits at infinity

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

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0⁺

One-sided limits

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

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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.

\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.

\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.

\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

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Squeeze theorem

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

\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\'Hôpital\'s rule

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

\displaystyle\lim_{x \to a}\dfrac{f(x)}{g(x)} = \lim_{x \to a}\dfrac{f\'(x)}{g\'(x)}

Continuity

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

\displaystyle f \text{ continuous at } a \iff \lim_{x \to a} f(x) = f(a)