Understanding the Limit of a Function

The limit of a function quantifies the behavior of the function as its input approaches a particular value. It establishes the foundational formalism for continuity, derivatives, and integrals in calculus by specifying the value approached by the function near a given point.

Mathematical Definition of the Limit of a Function at a Point

The formal definition of the limit of a function at a point uses the $\varepsilon$-$\delta$ criterion.

Let $f : D \to \mathbb{R}$ and $c$ be a limit point of $D \subseteq \mathbb{R}$. The real number $L$ is the limit of $f(x)$ as $x$ tends to $c$ if for each $\varepsilon > 0$, there exists $\delta > 0$ such that for all $x \in D$ with $0 < |x - c| < \delta$, we have $|f(x) - L| < \varepsilon$.

The notation for this is $\displaystyle\lim_{x \to c} f(x) = L$.

If no such $L$ exists, the limit does not exist at $x = c$.

Algebraic Properties and Laws for Limits at a Point

The algebraic rules for limits describe the behavior under addition, subtraction, multiplication, and division.

Suppose $\displaystyle\lim_{x \to a} f(x) = A$ and $\displaystyle\lim_{x \to a} g(x) = B$ exist. Then:

$\displaystyle\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) = A + B$

$\displaystyle\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) = A - B$

$\displaystyle\lim_{x \to a} [k f(x)] = k \cdot \lim_{x \to a} f(x) = kA$, where $k \in \mathbb{R}$

$\displaystyle\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) = AB$

$\displaystyle\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{A}{B}$, provided $B \neq 0$

For composite functions, if $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $\displaystyle\lim_{x \to a} f(g(x)) = f\left( \lim_{x \to a} g(x) \right)$.

Algebra Of Limits provides additional structured approaches to these rules.

Left-Hand and Right-Hand Limits at a Real Point

The left-hand limit is the value approached as $x \to a^-$, while the right-hand limit is the value as $x \to a^+$.

The left-hand limit at $a$ is defined as $\displaystyle\lim_{x \to a^-} f(x)$. The right-hand limit at $a$ is $\displaystyle\lim_{x \to a^+} f(x)$.

$\displaystyle\lim_{x \to a} f(x)$ exists if and only if $\displaystyle\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$ for some $L \in \mathbb{R}$.

If $\displaystyle\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$, the two-sided limit at $x = a$ does not exist.

Understanding Limit Of A Function includes graphical exploration of left and right limits.

Criteria for the Existence of the Limit of a Function

A function $f(x)$ has limit $L$ at $x = a$ if and only if both left-hand and right-hand limits exist and are equal to $L$.

If the left and right limits differ, the function is not continuous at $x = a$ and its general limit does not exist there.

Limit of a Function and Continuity

A function $f$ is continuous at $x = a$ if and only if all three of the following equality conditions hold:

$\displaystyle\lim_{x \to a} f(x) = f(a)$ and $f(a)$ is defined.

In terms of one-sided limits, continuity at $x = a$ requires $\displaystyle\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$.

When the limit exists but is not equal to $f(a)$, a removable discontinuity occurs at $x = a$.

For deeper study, see Limit, Continuity And Differentiability.

Standard Limits for Elementary and Trigonometric Functions

The following are rigorously established standard limits:

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

$\displaystyle\lim_{x \to 0} \frac{\tan x}{x}=1$

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

$\displaystyle\lim_{x \to 0} \frac{a^x-1}{x} = \ln a$, where $a>0$

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

These limits serve as references for solving indeterminate forms through substitution or manipulation.

Indeterminate Forms in Limits

The expressions $0/0$, $\infty/\infty$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^\infty$, and $\infty^0$ are called indeterminate forms and require algebraic or analytic manipulation to resolve.

L'Hospital's Rule is a systematic method for certain indeterminate forms, as detailed at L'Hospital's Rule For Indeterminate Limits.

Limit of a Function at Infinity and at a Point

The limit at infinity, $\displaystyle\lim_{x \to \infty} f(x)$, describes the end behavior. For rational functions, this can be analyzed by dividing numerator and denominator by the highest power of $x$ present.

The limit at a finite point is determined by examining behavior arbitrarily close to that point, as outlined previously.

Stepwise Examples of Finding Limits

Example 1: Evaluate $\displaystyle\lim_{x \to 2}\frac{x-2}{x^2-4}$.

Substitute $x = 2$ directly to obtain $0/0$, an indeterminate form.

Express denominator: $x^2-4 = (x-2)(x+2)$.

Rewrite as $\displaystyle\frac{x-2}{(x-2)(x+2)} = \frac{1}{x+2}$ for $x \neq 2$.

Take the limit: $\displaystyle\lim_{x \to 2}\frac{1}{x+2} = \frac{1}{4}$.

Solution: The value is $\frac{1}{4}$.

Solved Examples On Limits offers further stepwise solutions.

Example 2: Evaluate $\displaystyle\lim_{x \to 0}\frac{\sin(3x)}{x}$.

Express as $3\cdot \displaystyle\lim_{x \to 0}\frac{\sin(3x)}{3x}\cdot \frac{3x}{x} = 3\cdot 1 = 3$.

Solution: The value is $3$.

Example 3: Evaluate $\displaystyle\lim_{x \to 0}\frac{e^{2x} - 1}{x}$.

Apply the limit: $2 \cdot \displaystyle\lim_{x \to 0}\frac{e^{2x} - 1}{2x}$. Recognize $\displaystyle\lim_{y \to 0} \frac{e^{y} - 1}{y} = 1$, set $y=2x$.

Then, $2 \cdot 1 = 2$.

Solution: The value is $2$.

Example 4: Evaluate $\displaystyle\lim_{x \to 1} \frac{x^3-1}{x-1}$.

Factor numerator: $x^3-1 = (x-1)(x^2 + x + 1)$.

Write as $\displaystyle\frac{(x-1)(x^2 + x + 1)}{x-1} = x^2 + x + 1$ for $x \neq 1$.

Take limit: $1^2 + 1 + 1 = 3$.

Solution: The value is $3$.

Limit of a Function of Two Variables

For functions $f(x, y)$, the limit as $(x, y) \to (a, b)$ is $L$ if $|f(x, y) - L|$ can be made arbitrarily small whenever the distance $\sqrt{(x-a)^2 + (y-b)^2}$ is sufficiently small (but not zero).

The existence of the limit requires the value approached along every possible path towards $(a, b)$ to be the same.

For further problem sets on functions of several variables, see Mock Test On Limit And Continuity.