What are common limits?

Common limits are the limit identities used most often in calculus: limits of constants and powers, the foundational sin(x)/x limit and its trigonometric relatives, the limits that define e, the behavior of polynomials, exponentials, and logarithms at infinity, and one-sided limits where the two-sided limit fails to exist.

What is the limit of sin x over x as x approaches 0?

The limit of sin(x)/x as x approaches 0 equals 1. This is the foundational trigonometric limit, proven by the squeeze theorem using a geometric argument with the unit circle. Every standard derivative of a trigonometric function reduces to this identity.

What is the definition of e as a limit?

The constant e can be defined as a limit in two equivalent forms: e equals the limit of (1 + 1/x) to the power x as x approaches infinity, and e equals the limit of (1 + x) to the power 1/x as x approaches 0. Substituting u equal to 1 over x converts one form into the other.

What is L Hopital's rule?

L Hopital's rule resolves indeterminate forms of type 0 over 0 or infinity over infinity by differentiating the numerator and denominator separately. If the resulting limit of f-prime(x) over g-prime(x) exists, then the original limit equals it. The rule can be applied repeatedly when the result is still indeterminate.

What is the squeeze theorem?

The squeeze theorem says that if f(x) is less than or equal to g(x), which is less than or equal to h(x) near a point a, and the outer functions f and h both have the same limit L at a, then g must also have the limit L at a. It is the standard way to evaluate limits of bounded oscillating expressions, including sin(x)/x at 0 and at infinity.

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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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)