PracticeBack to top

Pomodoro

Pomodoro timer is idle

Limits

GitHub Changelog

Suggest edit Practice

1. What a limit means

A limit describes the value a function approaches as the input approaches some point.

The key idea is behavior near a point, not necessarily the function value at the point itself.

$$ \lim_{x \to a} f(x) = L $$

This reads as: "as $x$ gets close to $a$, $f(x)$ gets close to $L$."

Limits are the foundation of:

  • Derivatives

  • Continuity

  • Infinite series

  • Definite integrals

Intuition

If values of $f(x)$ near $x=a$ can be made arbitrarily close to $L$, then $L$ is the limit.

That means:

  • $f(a)$ may equal $L$

  • $f(a)$ may be different from $L$

  • $f(a)$ may not exist at all

Example:

$$ f(x) = \frac{x^2-9}{x-3} $$

For $x \ne 3$, this simplifies to:

$$ f(x) = x+3 $$

So:

$$ \lim_{x \to 3} \frac{x^2-9}{x-3} = 6 $$

even though the original formula is undefined at $x=3$.

Interactive visual

Approaching a limit

Move the removable discontinuity to see how the function approaches the same value from both sides.

Limit 2.0
Left/right values approach the same height

2. Formal definition

The rigorous definition of a limit uses $\varepsilon$ and $\delta$.

$$ \lim_{x \to a} f(x) = L $$

means:

for every $\varepsilon > 0$, there exists a $\delta > 0$ such that whenever

$$ 0 < |x-a| < \delta $$

it follows that

$$ |f(x)-L| < \varepsilon $$

How to read it

  • $\varepsilon$ is the allowed output error

  • $\delta$ is the required input closeness

  • The condition $0 < |x-a|$ excludes the point itself

This definition says that you can force the output to stay inside any desired tolerance by keeping the input sufficiently close to $a$.

Why the point is excluded

The value at the point does not control the limit.

For limits, only nearby values matter.

Common proof pattern

To prove a limit with $\varepsilon$-$\delta$:

  1. Start with $|f(x)-L|$

  2. Rewrite it in terms of $|x-a|$

  3. Bound any extra factors near $x=a$

  4. Choose a $\delta$ that makes the inequality work

This is usually easier once the function has been algebraically simplified.

3. Limit laws

If the individual limits exist, you can combine them algebraically.

Let

$$ \lim_{x \to a} f(x) = L,\qquad \lim_{x \to a} g(x) = M $$

Then:

$$ \lim_{x \to a} [f(x) \pm g(x)] = L \pm M $$
$$ \lim_{x \to a} [c\,f(x)] = cL $$
$$ \lim_{x \to a} [f(x)g(x)] = LM $$
$$ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M} $$

provided $M \ne 0$.

Other useful laws

If $n$ is a positive integer:

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

If $n$ is a root and the expression stays defined:

$$ \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{L} $$

If $p$ is a polynomial, then:

$$ \lim_{x \to a} p(x) = p(a) $$

If $r(x) = \frac{p(x)}{q(x)}$ is a rational function and $q(a)\ne 0$:

$$ \lim_{x \to a} r(x) = r(a) $$

Direct substitution

For continuous expressions, the fastest method is usually substitution.

Example:

$$ \lim_{x \to 2} (x^3 - 5x + 1) = 8 - 10 + 1 = -1 $$

If direct substitution gives an undefined form such as $\frac{0}{0}$, use another technique.

4. One-sided and infinite limits

One-sided limits

A left-hand limit approaches from values less than $a$:

$$ \lim_{x \to a^-} f(x) $$

A right-hand limit approaches from values greater than $a$:

$$ \lim_{x \to a^+} f(x) $$

The two-sided limit exists only if both one-sided limits exist and are equal.

$$ \lim_{x \to a} f(x) = L \quad \text{iff} \quad \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L $$

Example:

$$ f(x)= \begin{cases} 1, & x<0 \\ 3, & x \ge 0 \end{cases} $$

Then:

$$ \lim_{x \to 0^-} f(x)=1,\qquad \lim_{x \to 0^+} f(x)=3 $$

So $\lim_{x \to 0} f(x)$ does not exist.

Infinite limits

If $f(x)$ grows without bound near $a$, we write:

$$ \lim_{x \to a} f(x) = \infty $$

or

$$ \lim_{x \to a} f(x) = -\infty $$

This does not mean the limit is a real number. It means the function becomes arbitrarily large in magnitude.

Example:

$$ \lim_{x \to 2^-} \frac{1}{x-2} = -\infty,\qquad \lim_{x \to 2^+} \frac{1}{x-2} = +\infty $$

Because the one-sided behaviors differ, there is no finite two-sided limit at $x=2$.

When a limit does not exist

A limit fails to exist when:

  • Left-hand and right-hand limits differ

  • The function oscillates endlessly near the point

  • The function grows without bound and no finite limit is intended

Oscillation example:

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

does not exist because the function keeps oscillating between $-1$ and $1$.

5. Algebraic techniques

When direct substitution fails, simplify first.

1. Factor and cancel

Use this for removable discontinuities and $\frac{0}{0}$ forms.

Example:

$$ \lim_{x \to 1} \frac{x^2-1}{x-1} $$

Factor the numerator:

$$ \frac{(x-1)(x+1)}{x-1} = x+1 $$

So the limit is:

$$ 2 $$

2. Rationalize

Useful when square roots create $\frac{0}{0}$.

Example:

$$ \lim_{x \to 4} \frac{\sqrt{x}-2}{x-4} $$

Multiply by the conjugate:

$$ \frac{\sqrt{x}-2}{x-4}\cdot\frac{\sqrt{x}+2}{\sqrt{x}+2} = \frac{1}{\sqrt{x}+2} $$

Thus:

$$ \lim_{x \to 4} \frac{\sqrt{x}-2}{x-4} = \frac{1}{4} $$

3. Combine fractions

If a difference of rational expressions produces $\frac{0}{0}$, use a common denominator.

4. Use identities

Trig identities often reduce expressions to a standard form.

Examples:

$$ \sin^2 x + \cos^2 x = 1 $$
$$ 1 - \cos x = 2\sin^2\left(\frac{x}{2}\right) $$

5. Compare dominant terms

For limits at infinity, divide by the highest power or identify the leading growth rate.

6. Squeeze theorem

If

$$ g(x) \le f(x) \le h(x) $$

and

$$ \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L $$

then

$$ \lim_{x \to a} f(x) = L $$

This is especially useful for oscillating functions.

Example:

$$ -|x| \le x\sin\left(\frac{1}{x}\right) \le |x| $$

Since both bounds go to $0$ as $x \to 0$,

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

6. Special limits to know cold

These are standard results used constantly in calculus.

Trigonometric limits

$$ \lim_{x \to 0} \frac{\sin x}{x} = 1 $$
$$ \lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2} $$

More generally, for a constant $a$:

$$ \lim_{x \to 0} \frac{\sin(ax)}{x} = a $$

Exponential and logarithmic limits

$$ \lim_{x \to 0} \frac{e^x-1}{x} = 1 $$
$$ \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1 $$

Compound interest limit

$$ \lim_{x \to 0} (1+x)^{1/x} = e $$

Equivalently:

$$ \lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n = e $$

Why these matter

These limits are the building blocks for derivative formulas involving trig, exponential, and logarithmic functions.

7. Continuity and discontinuities

A function is continuous at $a$ if all three conditions hold:

  1. $f(a)$ is defined

  2. $\lim_{x \to a} f(x)$ exists

  3. $\lim_{x \to a} f(x) = f(a)$

In symbols:

$$ \lim_{x \to a} f(x) = f(a) $$

Types of discontinuity

TypeDescriptionTypical fix
RemovableA hole; the limit exists but the function value is missing or wrongRedefine the value
JumpLeft and right limits exist but are differentUsually none
InfiniteFunction blows up near the pointUsually none
OscillatoryFunction oscillates indefinitely near the pointUsually none

Continuity facts

Polynomials are continuous everywhere.

Rational functions are continuous wherever their denominator is nonzero.

Combinations of continuous functions are continuous wherever they are defined.

If $f$ is continuous and $g$ is continuous, then:

  • $f+g$ is continuous

  • $fg$ is continuous

  • $\frac{f}{g}$ is continuous where $g\ne 0$

  • $f\circ g$ is continuous where both compositions make sense

Why continuity helps

If a function is continuous at $a$, then the limit is immediate:

$$ \lim_{x \to a} f(x)=f(a) $$

That is the fastest possible limit evaluation.

8. Limits at infinity and asymptotes

Limits at infinity describe end behavior.

Horizontal asymptotes

If

$$ \lim_{x \to \infty} f(x) = L $$

then $y=L$ is a horizontal asymptote.

If the same happens as $x \to -\infty$, that also gives a horizontal asymptote.

Example:

$$ \lim_{x \to \infty}\frac{3x^2-1}{2x^2+5}=\frac{3}{2} $$

So $y=\frac{3}{2}$ is a horizontal asymptote.

Rational functions at infinity

For

$$ \frac{p(x)}{q(x)} $$

compare degrees:

  • Degree of numerator < degree of denominator: limit is $0$

  • Equal degrees: limit is ratio of leading coefficients

  • Numerator degree > denominator degree: no finite horizontal asymptote

Vertical asymptotes

If $f(x)$ grows without bound near $x=a$, then $x=a$ may be a vertical asymptote.

Example:

$$ \lim_{x \to 2} \frac{1}{(x-2)^2} = +\infty $$

So $x=2$ is a vertical asymptote.

Oblique asymptotes

If a rational function has numerator degree exactly one higher than the denominator, polynomial division may reveal a slant asymptote.

9. Indeterminate forms and L'Hôpital's rule

An indeterminate form is an expression that does not determine the limit by itself.

Common forms:

  • $\frac{0}{0}$

  • $\frac{\infty}{\infty}$

  • $0\cdot\infty$

  • $\infty-\infty$

  • $0^0$

  • $1^\infty$

  • $\infty^0$

The form tells you that more work is needed; it does not tell you the answer.

L'Hôpital's rule

If

$$ \lim_{x \to a} f(x)=0,\qquad \lim_{x \to a} g(x)=0 $$

or both functions approach $\pm\infty$, and the derivatives exist near $a$, then:

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

when the right-hand limit exists and the rule's hypotheses are satisfied.

When to use it

Use L'Hôpital's rule after:

  • Confirming an indeterminate form

  • Checking that derivatives exist where needed

  • Trying simpler algebra first when possible

When not to overuse it

L'Hôpital's rule is powerful, but algebra is often faster.

For example, factorization or a standard trig limit is usually cleaner than repeated differentiation.

Example

$$ \lim_{x \to 0} \frac{e^x-1}{x} $$

This is $\frac{0}{0}$, so L'Hôpital gives:

$$ \lim_{x \to 0} \frac{e^x}{1} = 1 $$

10. Problem-solving workflow

When you see a limit, use this sequence.

  1. Try direct substitution.

  2. If you get an ordinary number, you are done.

  3. If you get $\frac{0}{0}$, simplify algebraically.

  4. If radicals appear, rationalize.

  5. If trig expressions appear, look for standard identities or special limits.

  6. If $x \to \infty$ or $x \to -\infty$, compare leading terms.

  7. If the function is piecewise, compute left and right limits separately.

  8. If the expression is oscillatory or complicated, consider the squeeze theorem.

  9. Use L'Hôpital's rule only when the hypotheses fit and a simpler approach is not better.

Common mistakes

  • Treating the limit as the same thing as the function value

  • Cancelling terms before checking for zero denominators

  • Forgetting to check both one-sided limits

  • Using L'Hôpital's rule on a non-indeterminate form

  • Replacing $x$ with $\infty$ as if it were a number

  • Ignoring domain restrictions

Fast diagnostic questions

  • Is the function continuous at the point?

  • Does substitution produce a valid value?

  • Is this a removable discontinuity?

  • Are you looking at behavior from one side only?

  • Is the variable heading toward infinity?

11. Formula sheet

Core notation

$$ \lim_{x \to a} f(x) = L $$
$$ \lim_{x \to a^-} f(x),\qquad \lim_{x \to a^+} f(x) $$

Limit laws

$$ \lim_{x \to a}[f(x)\pm g(x)] = L \pm M $$
$$ \lim_{x \to a}[f(x)g(x)] = LM $$
$$ \lim_{x \to a}\frac{f(x)}{g(x)} = \frac{L}{M}, \quad M\ne 0 $$

Standard limits

$$ \lim_{x \to 0}\frac{\sin x}{x}=1 $$
$$ \lim_{x \to 0}\frac{1-\cos x}{x^2}=\frac12 $$
$$ \lim_{x \to 0}\frac{e^x-1}{x}=1 $$
$$ \lim_{x \to 0}\frac{\ln(1+x)}{x}=1 $$
$$ \lim_{n \to \infty}\left(1+\frac1n\right)^n=e $$

Continuity test

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

Squeeze theorem

$$ g(x)\le f(x)\le h(x),\quad \lim_{x \to a} g(x)=\lim_{x \to a} h(x)=L $$

implies

$$ \lim_{x \to a} f(x)=L $$

L'Hôpital's rule

For $\frac{0}{0}$ or $\frac{\infty}{\infty}$ forms:

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

when the rule applies.

Sources