Limit Calculator

Understanding Limits

In calculus, a limit describes the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential because they are used to define continuity, derivatives, and integrals.

The notation $\lim_{x \to a} f(x) = L$ means that as $x$ gets arbitrarily close to $a$ (from both sides), the value of $f(x)$ gets arbitrarily close to $L$.

Types of Limits

1. Two-Sided Limit

For a standard limit $\lim_{x \to a} f(x)$ to exist, the function must approach the same value from both the left and the right sides.

2. One-Sided Limits

• Left-hand limit: $\lim_{x \to a^-} f(x)$ approaches $a$ from values less than $a$.
• Right-hand limit: $\lim_{x \to a^+} f(x)$ approaches $a$ from values greater than $a$.

If $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$, then the two-sided limit exists and equals $L$.

3. Limits at Infinity

Describes the behavior of a function as $x$ grows without bound ($\lim_{x \to \infty} f(x)$) or decreases without bound ($\lim_{x \to -\infty} f(x)$). These limits identify horizontal asymptotes.

Techniques for Evaluating Limits

Direct Substitution

The first step is always to try plugging in the value $a$ into the function. If $f(a)$ is defined and the function is continuous, that is the limit.

• Example: $\lim_{x \to 2} (3x + 1) = 3(2) + 1 = 7$

Factoring (Removing Indeterminacy)

If direct substitution results in $\frac{0}{0}$ (indeterminate form), try factoring and canceling common terms.

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

Conjugate Method

Useful for limits involving square roots that result in $\frac{0}{0}$. Multiply both numerator and denominator by the conjugate.

L'Hôpital's Rule

A powerful technique for indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. It states that the limit of the quotient of functions is equal to the limit of the quotient of their derivatives: $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$

Frequently Asked Questions

Q: What if the limit results in division by zero (e.g., 5/0)? A: If the numerator is non-zero and the denominator is zero, the limit does not exist (DNE). It typically indicates a vertical asymptote where the function goes to $+\infty$ or $-\infty$.

Q: Does a function have to be defined at a point for a limit to exist? A: No. A limit describes behavior near a point, not exactly at the point. For example, the function $\frac{x^2-1}{x-1}$ has a limit of 2 as $x \to 1$, even though it is undefined at $x=1$.

Q: What is an indeterminate form? A: Forms like $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \times \infty$, $\infty - \infty$ do not give a definitive answer. They require further manipulation (like factoring or L'Hôpital's rule) to evaluate.