Limit Formulas: Laws And Standard Limits With Examples

What Are The Limit Formulas?

Limit formulas are the standard results used to find $\lim_{x \to a} f(x)$, the value the output of $f$ approaches as $x$ moves toward $a$. They matter because the obvious move, substituting $x = a$, fails whenever the expression collapses to an indeterminate form like $\frac{0}{0}$. The formulas hand you the answer those cases hide.

There are two families, and every problem uses one or both.

• The limit laws. These break a complicated limit into simpler pieces: the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and so on. They work whenever the individual limits exist.

• The standard limits. These are the specific $0/0$ and $1^{\infty}$ results, memorised once and reused, such as $\lim_{x \to 0}\frac{\sin x}{x} = 1$ and $\lim_{x \to \infty}\left(1 + \frac{1}{x}\right)^{x} = e$.

A limit describes approach, not arrival. The function need not even be defined at $a$; what counts is the value the outputs cluster around as $x$ closes in. For the fuller picture of that idea, see limit of a function.

What Are The Limit Laws?

The limit laws let you compute the limit of a combination from the limits of the parts. Suppose $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$ both exist, and $k$ is a constant. Then:

$$\lim_{x \to a}\big[f(x) + g(x)\big] = L + M \qquad \text{(sum)}$$

$$\lim_{x \to a}\big[f(x) - g(x)\big] = L - M \qquad \text{(difference)}$$

$$\lim_{x \to a}\big[k \cdot f(x)\big] = kL \qquad \text{(constant multiple)}$$

$$\lim_{x \to a}\big[f(x),g(x)\big] = L \cdot M \qquad \text{(product)}$$

$$\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{L}{M}, \quad \text{provided } M \neq 0 \qquad \text{(quotient)}$$

$$\lim_{x \to a}\big[f(x)\big]^{n} = L^{n} \qquad \text{(power)}$$

The one condition worth circling is on the quotient law: the denominator's limit $M$ must not be zero. When it is, the law does not apply, and you have reached exactly the situation the standard limits below were built for. A full reference for these six rules and their proofs lives at limit laws.

What Are The Standard Limit Formulas?

These are the results to commit to memory. Each one resolves a form that direct substitution turns into $\frac{0}{0}$, $\frac{\infty}{\infty}$, or $1^{\infty}$. Every entry below was checked numerically against its closed form before being listed.

Table: The standard limit formulas, grouped by function type.

A few reading notes keep these accurate. In every trigonometric row the angle must be measured in radians, and the scope $x \to 0$ (or $x \to \infty$) is part of the formula, not decoration, since the same expression has a different limit elsewhere. The symbol $\ln a$ in the $a^{x}$ row is the natural logarithm, which is why the $e^{x}$ case, where $\ln e = 1$, comes out to exactly $1$. The last row is really the derivative of $x^{n}$ in disguise, a bridge explored in limits and derivatives.

How Do You Prove The Standard Limit sin x / x = 1?

The trigonometric limits all rest on one geometric argument for $\dfrac{\sin x}{x}$, and the tool is the squeeze theorem.

Take a small positive angle $x$ (in radians) on the unit circle. Comparing three areas, the triangle inside the sector, the sector itself, and the larger right triangle outside, gives the chain

$$\sin x ; < ; x ; < ; \tan x.$$

Divide every part by $\sin x$ (positive, so the inequalities hold):

$$1 ; < ; \frac{x}{\sin x} ; < ; \frac{1}{\cos x}.$$

Take reciprocals, which flips the direction:

$$\cos x ; < ; \frac{\sin x}{x} ; < ; 1.$$

As $x \to 0$, the left side $\cos x \to 1$ and the right side is already $1$. The ratio $\frac{\sin x}{x}$ is squeezed between two quantities both heading to $1$, so it has nowhere else to go:

$$\lim_{x \to 0}\frac{\sin x}{x} = 1.$$

The result for $\dfrac{\tan x}{x}$ follows at once, since $\tan x = \dfrac{\sin x}{\cos x}$, giving $\dfrac{\tan x}{x} = \dfrac{\sin x}{x}\cdot\dfrac{1}{\cos x} \to 1 \cdot 1 = 1$.

How Do You Use The Limit Formulas? Worked Examples

Each example names the formula it leans on, then applies it step by step. Every answer was verified numerically.

Example 1: A trigonometric limit where the angle is not $x$.

Evaluate $\displaystyle\lim_{x \to 0}\frac{\sin 5x}{x}$.

Direct substitution gives $\frac{0}{0}$, so reshape the expression to match $\frac{\sin(\text{angle})}{\text{angle}}$. Multiply and divide by $5$:

$$\lim_{x \to 0}\frac{\sin 5x}{x} = \lim_{x \to 0}\left(\frac{\sin 5x}{5x}\cdot 5\right) = 5 \cdot \lim_{x \to 0}\frac{\sin 5x}{5x} = 5 \cdot 1 = 5$$

As $x \to 0$, the inner angle $5x \to 0$ too, so $\frac{\sin 5x}{5x} \to 1$.

Final answer: $\displaystyle\lim_{x \to 0}\frac{\sin 5x}{x} = 5$.

Example 2: A quotient of two trigonometric limits.

Evaluate $\displaystyle\lim_{x \to 0}\frac{\tan 3x}{\sin 2x}$.

Rewrite each factor against its own angle so the standard limits appear:

$$\frac{\tan 3x}{\sin 2x} = \frac{\tan 3x}{3x}\cdot\frac{2x}{\sin 2x}\cdot\frac{3x}{2x} = \frac{\tan 3x}{3x}\cdot\frac{2x}{\sin 2x}\cdot\frac{3}{2}$$

Each of the first two factors tends to $1$ as $x \to 0$, leaving

$$\lim_{x \to 0}\frac{\tan 3x}{\sin 2x} = 1 \cdot 1 \cdot \frac{3}{2} = \frac{3}{2}.$$

Final answer: $\displaystyle\lim_{x \to 0}\frac{\tan 3x}{\sin 2x} = \frac{3}{2}$.

Example 3: An exponential limit.

Evaluate $\displaystyle\lim_{x \to 0}\frac{e^{3x} - 1}{x}$.

Match the standard form $\frac{e^{u} - 1}{u}$ by writing the exponent as the denominator:

$$\lim_{x \to 0}\frac{e^{3x} - 1}{x} = \lim_{x \to 0}\left(\frac{e^{3x} - 1}{3x}\cdot 3\right) = 3 \cdot 1 = 3$$

Final answer: $\displaystyle\lim_{x \to 0}\frac{e^{3x} - 1}{x} = 3$.

Example 4: The algebraic power formula.

Evaluate $\displaystyle\lim_{x \to 2}\frac{x^{5} - 32}{x - 2}$.

This is the form $\frac{x^{n} - a^{n}}{x - a}$ with $a = 2$ and $n = 5$, since $32 = 2^{5}$. Apply $n,a^{,n-1}$:

$$\lim_{x \to 2}\frac{x^{5} - 32}{x - 2} = 5 \cdot 2^{,4} = 5 \cdot 16 = 80$$

Final answer: $\displaystyle\lim_{x \to 2}\frac{x^{5} - 32}{x - 2} = 80$.

What Is L'Hopital's Rule, And When Does It Apply?

Some limits do not match any standard formula. For those, L'Hopital's rule is the general tool for the two indeterminate quotients. If $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ both give $0$, or both give $\pm\infty$, so the ratio is $\frac{0}{0}$ or $\frac{\infty}{\infty}$, and the functions are differentiable near $a$, then

$$\lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)},$$

where $f'$ and $g'$ are the derivatives. You differentiate the top and bottom separately, not as a quotient, then try the limit again.

Two cautions keep it honest. First, check the form before you use it; applying L'Hopital to a limit that is not $\frac{0}{0}$ or $\frac{\infty}{\infty}$ gives wrong answers. Second, the standard limits already settle the common $0/0$ cases faster, so reach for L'Hopital only when no formula fits. For the full catalogue of forms it handles, see indeterminate forms.

Why Do The Limit Formulas Work?

The formulas are not arbitrary. Each one reflects a fact about how numbers behave under approach.

• The laws inherit ordinary arithmetic. A limit is the single value that outputs crowd toward. If $f$'s outputs settle on $L$ and $g$'s settle on $M$, then their sum settles on $L + M$ and their product on $L \cdot M$, for the same reason that adding two numbers near $L$ and $M$ lands near $L + M$. The laws simply carry the algebra of numbers across the "approaches" bridge.

• The trigonometric limit is geometry. $\frac{\sin x}{x} \to 1$ says that for a tiny angle, the arc and its sine are nearly equal in length. Zoom far enough into a circle and it looks straight, so the curved arc $x$ and the vertical drop $\sin x$ become indistinguishable.

• The number $e$ is compounding taken to the limit. $\left(1 + \frac{1}{x}\right)^{x} \to e$ is what happens when interest is compounded not yearly or daily but continuously. The base $e$ is defined by that very limit, which is why it appears the moment growth becomes smooth.

Seen this way, the standard limits are less a list to memorise than three ideas, geometry, continuity of arithmetic, and continuous growth, wearing formula clothing.

Who Invented The Modern Idea Of A Limit?

For over a century calculus worked without a precise definition of the very thing it rested on. The fix came from two mathematicians who insisted on rigour.

Two more figures sit behind the number $e$:

• Jacob Bernoulli (1655–1705, Switzerland) met the limit $\left(1 + \frac{1}{x}\right)^{x}$ while studying continuously compounded interest, and showed it settles on a value between $2$ and $3$.

• Leonhard Euler (1707–1783, Switzerland) named that value $e$, computed it to many places, and placed it at the centre of exponential and logarithmic analysis.

Where Are The Limit Formulas Used In The Real World?

Because a limit captures "the value something approaches", the formulas appear wherever a rate, a total, or a smooth process is modelled.

• Physics and motion: instantaneous velocity is a limit of average velocity over shrinking time intervals, computed with the $\frac{x^{n} - a^{n}}{x - a}$ formula when position is a power of time.

• Finance: continuously compounded interest uses $\left(1 + \frac{1}{x}\right)^{x} \to e$ directly, turning a discrete compounding schedule into a smooth exponential.

• Engineering and signals: the small-angle result $\frac{\sin x}{x} \to 1$ justifies replacing $\sin x$ with $x$ in pendulum, optics, and vibration models, which linearises otherwise hard equations.

• Computer graphics: anti-aliasing and shading rely on the $\frac{\sin x}{x}$ function (the "sinc" curve), whose behaviour near zero is exactly this limit.

• Probability and growth models: populations, radioactive decay, and cooling all reduce to $e$, defined by the limit in the row above.

One short table of limits quietly underwrites motion, money, waves, and growth, which is why every calculus course opens with it.

What Are The Most Common Mistakes With The Limit Formulas?

These three errors account for most lost marks on limits, and each matches a question real students ask on r/calculus, r/learnmath, and course handouts on indeterminate forms.

Plugging in the value on an indeterminate form.

Where it slips in:

Faced with $\lim_{x \to 0}\frac{\sin x}{x}$, a student substitutes $x = 0$, gets $\frac{0}{0}$, and writes the answer as $0$ or as "undefined".

Don't do this:

Do not treat $\frac{0}{0}$ as a number. It is an indeterminate form, a signal that more work is needed, not a result.

The correct way:

Recognise the form, then apply the matching standard limit or L'Hopital's rule. Here $\frac{\sin x}{x} \to 1$, not $0$.

Dropping the $x \to 0$ scope.

Where it slips in:

A student quotes $\frac{\sin x}{x} = 1$ as if it held for every $x$, then uses it inside a limit as $x \to \infty$ or at some other point.

Don't do this:

Do not detach a standard limit from the arrow it was proved under. $\frac{\sin x}{x}$ equals $1$ only in the limit $x \to 0$; as $x \to \infty$ the same ratio tends to $0$.

The correct way:

Carry the $\lim_{x \to 0}$ with the formula every time, and confirm the variable is actually heading to the point the formula requires.

Ignoring the angle when it is not $x$.

Where it slips in:

A student writes $\lim_{x \to 0}\frac{\sin 5x}{x} = 1$, matching the shape by eye without checking that the angle and the denominator agree.

Don't do this:

Do not apply $\frac{\sin(\text{angle})}{\text{angle}} = 1$ unless the denominator is the same angle that sits inside the sine.

The correct way:

Force the match. Rewrite $\frac{\sin 5x}{x}$ as $\frac{\sin 5x}{5x}\cdot 5$, so the ratio is $\frac{\sin 5x}{5x} \to 1$ and the answer is $5$.

Practice Problems On The Limit Formulas

Work each one, then check against the answer. Answers are verified.

1. Evaluate $\displaystyle\lim_{x \to 0}\frac{\sin 7x}{\sin 3x}$.
   (Answer: $\frac{7}{3}$.)

2. Evaluate $\displaystyle\lim_{x \to 0}\frac{\ln(1 + 2x)}{x}$.
   (Answer: $2$.)

3. Evaluate $\displaystyle\lim_{x \to 0}\frac{5^{x} - 1}{x}$.
   (Answer: $\ln 5 \approx 1.6094$.)

4. Evaluate $\displaystyle\lim_{x \to 2}\frac{x^{3} - 8}{x - 2}$.
   (Answer: $3 \cdot 2^{2} = 12$.)

5. Evaluate $\displaystyle\lim_{x \to 0}\frac{1 - \cos 2x}{x}$.
   (Answer: $0$.)

6. Evaluate $\displaystyle\lim_{x \to \infty}\left(1 + \frac{1}{x}\right)^{2x}$.
   (Answer: $e^{2} \approx 7.3891$.)

Where Should You Go Next After Limit Formulas?

The formulas open several natural doors into calculus.

1. Standard limits. Drill the memorised results on their own, with more worked cases for each row of the table.

2. Limits at infinity. See how the same idea behaves as $x \to \infty$, where growth rates decide the answer.

3. Continuity of a function. A function is continuous exactly when its limit equals its value, so this is the next concept limits unlock.

If your child is meeting limits for the first time, a live Bhanzu trainer teaches the formulas from the "value it approaches" picture up, so the standard limits feel earned rather than memorised, in the Bhanzu math program.