Limit Calculator

A limit describes the value that a function approaches as the input approaches a particular point. The formal definition states:

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

means that for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$.

Intuitively, a limit answers: "What value does $f(x)$ get arbitrarily close to as $x$ gets close to $a$?"

One-sided limits approach from a single direction:

• Left-hand limit: $\lim_{x \to a^-} f(x)$
• Right-hand limit: $\lim_{x \to a^+} f(x)$

A two-sided limit exists only when both one-sided limits exist and are equal.

Limits at infinity describe end behavior:

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

means $f(x)$ approaches $L$ as $x$ grows without bound.

Limits are foundational to calculus — they define derivatives, integrals, and continuity. A function is continuous at $a$ if and only if $\lim_{x \to a} f(x) = f(a)$.

Method 1: Direct Substitution

The simplest approach — plug in the value. If $f(a)$ is defined and the function is continuous at $a$:

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

Example: $\lim_{x \to 3} (x^2 + 1) = 9 + 1 = 10$

Method 2: Factoring and Cancellation

When direct substitution yields $\frac{0}{0}$, factor and cancel:

$\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2}(x+2) = 4$

Method 3: L'Hôpital's Rule

When direct substitution gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$:

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

provided the right-hand limit exists.

Example: $\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1$

Method 4: Squeeze Theorem

If $g(x) \leq f(x) \leq h(x)$ near $a$, and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

Method 5: Conjugate Multiplication

For expressions with radicals:

$\lim_{x \to 0} \frac{\sqrt{x+4} - 2}{x} = \lim_{x \to 0} \frac{(\sqrt{x+4}-2)(\sqrt{x+4}+2)}{x(\sqrt{x+4}+2)} = \lim_{x \to 0} \frac{x}{x(\sqrt{x+4}+2)} = \frac{1}{4}$

Important Standard Limits

Comparison of Methods

• Applying L'Hôpital's Rule without verifying indeterminate form: The rule only applies to $\frac{0}{0}$ or $\frac{\infty}{\infty}$. Using it on $\frac{1}{0}$ or other forms gives wrong answers.
• Confusing limit existence with function value: $\lim_{x \to a} f(x)$ can exist even if $f(a)$ is undefined. The limit depends on nearby values, not the value at the point.
• Ignoring one-sided limits: For piecewise functions or at discontinuities, always check both left and right limits separately.
• Incorrectly distributing limits over indeterminate arithmetic: $\lim(f - g) \neq \lim f - \lim g$ when both are $\infty$ (gives $\infty - \infty$, which is indeterminate).
• Treating $\frac{\infty}{\infty}$ as 1: $\frac{\infty}{\infty}$ is indeterminate — it can equal any value.