Limit Formulas and Laws with Conditions and PDF

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Limit laws

Combine component limits when each required limit exists and every denominator stays nonzero.

$\lim_{x\to a}c=c$: Constant limit lawc is constant with respect to x.
Use this law for a term that does not depend on the approaching variable.
$\lim_{x\to a}[f(x)+g(x)]=L+M$: Limit sum lawThe limit of f of x as x approaches a equals L. The limit of g of x as x approaches a equals M. L and M are finite real numbers for the form shown.
Use this law to split the limit of a sum when both component limits exist.
$\lim_{x\to a}[f(x)g(x)]=LM$: Limit product lawThe limit of f of x as x approaches a equals L. The limit of g of x as x approaches a equals M. L and M are finite real numbers for the form shown.
Use this law to multiply two component limits that both exist.
$\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{L}{M}$: Limit quotient lawThe limit of f of x as x approaches a equals L. The limit of g of x as x approaches a equals M. M is not zero.
Use this law to divide two component limits when the denominator limit is nonzero.
$\lim_{x\to a}[f(x)]^n=L^n$: Power limit lawThe limit of f of x as x approaches a equals L. The power operation is real-valued and continuous at L.
Use this law after establishing that f approaches L and the real-valued power is continuous at L.

Continuity and composition

Pass a limit through a continuous outer function after establishing the inner limit.

$\lim_{x\to a}g(f(x))=g(L)$: Continuous composition lawThe limit of f of x as x approaches a equals L. g is continuous at L.
Use this law when f approaches L and the outer function g is continuous at L.

Standard limits

Recognize the small-angle and exponential patterns that anchor many indeterminate limits.

$\lim_{x\to 0}\frac{\sin x}{x}=1$: Standard sine limitAngles are measured in radians. The limit is two-sided over the real numbers.
Use this standard limit when a trigonometric expression reduces to sine of a small angle divided by that angle.
$\lim_{x\to0}\frac{1-\cos x}{x}=0$: Cosine difference limitAngles are measured in radians. The limit is two-sided over the real numbers.
Use this standard limit for a first-power small-angle denominator; related squared-denominator forms have a different value.
$\lim_{x\to0}\frac{e^x-1}{x}=1$: Standard exponential limitThe base is the natural exponential e. The limit is two-sided over the real numbers.
Use this limit when an exponential difference has a matching small increment in the denominator.
$\lim_{x\to0}\frac{\ln(1+x)}{x}=1$: Standard logarithm limitx is greater than negative one near the approach point. The logarithm is natural logarithm.
Use this limit when a natural logarithm approaches log of one and the denominator matches its small increment.

Rational end behavior

Compare leading powers when a rational function approaches positive or negative infinity.

$\lim_{x\to\pm\infty}\frac{a_nx^n+\cdots}{b_nx^n+\cdots}=\frac{a_n}{b_n}$: Equal-degree rational end behaviora sub n and b sub n are the nonzero leading coefficients. The numerator and denominator have the same finite degree n.
Use this rule for polynomial ratios whose numerator and denominator have the same highest degree.

Notation

Symbols used on this sheet

Symbol	Meaning	Usage note
$x\to a$	x approaches a	The input can approach without equaling the point.
$x\to a^-$	Left-hand approach	Only values less than a are used.
$x\to a^+$	Right-hand approach	Only values greater than a are used.
$L,M$	Component limit values	The finite forms shown assume these component limits exist.
$M\ne0$	Nonzero denominator limit	Required by the displayed quotient law.
$\sin x$	Sine of a radian angle	The standard sine limit equals one only with radian measure.

Use the reference

Short applications

These examples show when to choose a formula; detailed instruction belongs in the linked guide or calculator.

Remove a radical indeterminate form

Rationalization converts the zero-over-zero expression to a reciprocal that is continuous at the approach point.

\lim_{x\to4}\frac{\sqrt{x+5}-3}{x-4}

\lim_{x\to4}\frac{1}{\sqrt{x+5}+3}=\frac16

Condition check: The result also equals the derivative of square root x plus 5 at x = 4.

Rescale the standard sine limit

Multiplying and dividing by 5x exposes sine u divided by u.

\lim_{x\to0}\frac{\sin(5x)}x

5\lim_{x\to0}\frac{\sin(5x)}{5x}=5

Condition check: Symmetric high-precision samples near zero approach 5 from both sides.

Use conditions, not memory alone

Common formula confusions

Treating an indeterminate form as an answer

Reporting zero over zero after direct substitution.

Fix: Use the form to choose factoring, rationalization, a standard limit, squeezing, or another justified method.

Ignoring one-sided disagreement

Reporting a two-sided limit when the left and right limits differ.

Fix: Compute both one-sided limits at jumps, corners, absolute values, and vertical asymptotes.

Dividing by a zero limit

Applying the quotient law when the denominator limit is zero.

Fix: Analyze the resulting form instead of using a law whose nonzero condition fails.

Using degrees in a standard trigonometric limit

Applying the value one to sine x over x with x measured in degrees.

Fix: Convert to radians or include the degree-to-radian scale factor explicitly.

Common questions

Limit formulas and laws FAQ

Which limit formulas are included?

The focused sheet includes the sum, product, and quotient laws plus the standard sine limit, with existence, nonzero-denominator, and radian-measure conditions.

Can I substitute the approach value into every limit?

Direct substitution works for functions continuous at the point. An indeterminate form or discontinuity requires more analysis.

What is the difference between a one-sided and two-sided limit?

A one-sided limit follows inputs from only one direction. A two-sided limit exists only when the left-hand and right-hand limits both exist and agree.

Does the limit PDF match the web reference?

Yes. Both formats use the same structured formula records, including the hypotheses and accessible descriptions.

Sources and curriculum alignment

This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.

2.3 The Limit Laws, OpenStax Calculus Volume 1
2.2 The Limit of a Function, OpenStax Calculus Volume 1

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