Understanding Limits in Calculus: A Comprehensive Guide, Limit Rules and Examples

In calculus, the concept of a limit is fundamental. It describes the behavior of a function as its input approaches a particular value. Limits are crucial for defining continuity, derivatives, and integrals.

What is a Limit?

Intuitive Idea:
Imagine you’re walking along the graph of a function, $f(x)$ . As your x-coordinate gets closer and closer to a certain number, say $c$ , the limit is the y-coordinate that your path is heading towards. Importantly, the limit doesn’t care what the function’s value actually is at $x=c$ (or even if it’s defined there), only what value it approaches.

Notation:
We write the limit of a function $f(x)$ as $x$ approaches $c$ is $L$ as:
$\lim_{x \to c} f(x) = L$
This is read as “the limit of $f(x)$ as $x$ approaches $c$ equals $L$ .”

One-Sided Limits:
Sometimes, the behavior of a function as $x$ approaches $c$ can be different from the left side versus the right side.
Left-hand limit: $\lim_{x \to c^-} f(x) = L_1$ means $f(x)$ approaches $L_1$ as $x$ approaches $c$ from values less than $c$ .
Right-hand limit: $\lim_{x \to c^+} f(x) = L_2$ means $f(x)$ approaches $L_2$ as $x$ approaches $c$ from values greater than $c$ .

For the overall limit to exist (i.e., $\lim_{x \to c} f(x) = L$ ), the left-hand limit and the right-hand limit must both exist and be equal to $L$ .
$\lim_{x \to c} f(x) = L \quad \iff \quad \lim_{x \to c^-} f(x) = L \quad \text{and} \quad \lim_{x \to c^+} f(x) = L$

When does a limit not exist?
A limit might not exist at a point $x=c$ if:

The left-hand and right-hand limits are different (a “jump” in the function).
The function approaches $\infty$ or $-\infty$ from one or both sides (an infinite discontinuity or vertical asymptote).
The function oscillates infinitely as $x$ approaches $c$ (e.g., $\lim_{x \to 0} \sin(1/x)$ ).

Formal Definition (Epsilon-Delta Definition):
While the intuitive idea is helpful, the precise mathematical definition is the epsilon-delta ( $\epsilon-\delta$ ) definition. It states:
Let $f(x)$ be a function defined on an open interval containing $c$ (except possibly at $c$ itself) and let $L$ be a real number. The statement $\lim_{x \to c} f(x) = L$ means that for every $\epsilon > 0$ , there exists a $\delta > 0$ such that if $0 < |x - c| < \delta$ , then $|f(x) - L| < \epsilon$ .

In simpler terms: you can make $f(x)$ as close as you want to $L$ (within a distance $\epsilon$ ) by choosing $x$ values sufficiently close to $c$ (within a distance $\delta$ , but not equal to $c$ ).

Limit Rules (Properties of Limits)

These rules allow us to calculate limits of complex functions by breaking them down into simpler parts. Assume that $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$ , where $L$ and $M$ are real numbers.

Limit of a Constant:
If $k$ is a constant, then
$\lim_{x \to c} k = k$
Example: $\lim_{x \to 5} 10 = 10$

Limit of $x$ :
$\lim_{x \to c} x = c$
Example: $\lim_{x \to 3} x = 3$

Constant Multiple Rule:
If $k$ is a constant, then
$\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x) = k \cdot L$
Example: If $\lim_{x \to 2} f(x) = 5$ , then $\lim_{x \to 2} [3 f(x)] = 3 \cdot 5 = 15$

Sum Rule:
$\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x) = L + M$
Example: If $\lim_{x \to 1} f(x) = 2$ and $\lim_{x \to 1} g(x) = 7$ , then $\lim_{x \to 1} [f(x) + g(x)] = 2 + 7 = 9$

Difference Rule:
$\lim_{x \to c} [f(x) - g(x)] = \lim_{x \to c} f(x) - \lim_{x \to c} g(x) = L - M$
Example: If $\lim_{x \to 1} f(x) = 2$ and $\lim_{x \to 1} g(x) = 7$ , then $\lim_{x \to 1} [f(x) - g(x)] = 2 - 7 = -5$

Product Rule:
$\lim_{x \to c} [f(x) \cdot g(x)] = \left(\lim_{x \to c} f(x)\right) \cdot \left(\lim_{x \to c} g(x)\right) = L \cdot M$
Example: If $\lim_{x \to 4} f(x) = 3$ and $\lim_{x \to 4} g(x) = -2$ , then $\lim_{x \to 4} [f(x) \cdot g(x)] = 3 \cdot (-2) = -6$

Quotient Rule:
Provided that $\lim_{x \to c} g(x) = M \neq 0$ ,
$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} = \frac{L}{M}$
Example: If $\lim_{x \to 0} f(x) = 6$ and $\lim_{x \to 0} g(x) = 2$ , then $\lim_{x \to 0} \frac{f(x)}{g(x)} = \frac{6}{2} = 3$
(If $M=0$ and $L \neq 0$ , the limit will be $\infty$ , $-\infty$ , or DNE. If $M=0$ and $L=0$ , this is an indeterminate form, and other techniques like factoring, rationalizing, or L’Hôpital’s Rule are needed.)

Power Rule:
If $n$ is a positive integer,
$\lim_{x \to c} [f(x)]^n = \left[\lim_{x \to c} f(x)\right]^n = L^n$
This also holds for rational exponents $n = p/q$ provided $L^{p/q}$ is a real number.
Example: If $\lim_{x \to 2} f(x) = 3$ , then $\lim_{x \to 2} [f(x)]^4 = 3^4 = 81$

Root Rule:
If $n$ is a positive integer,
$\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to c} f(x)} = \sqrt[n]{L}$
(If $n$ is even, we must have $L \ge 0$ .)
Example: If $\lim_{x \to 5} f(x) = 16$ , then $\lim_{x \to 5} \sqrt{f(x)} = \sqrt{16} = 4$

Direct Substitution Property

For many well-behaved functions (called continuous functions at $x=c$ ), the limit at a point $c$ can be found by simply substituting $c$ into the function:
$\lim_{x \to c} f(x) = f(c)$
This property holds for:
Polynomial functions: If $p(x)$ is a polynomial, then $\lim_{x \to c} p(x) = p(c)$ .
Example: $\lim_{x \to 2} (x^2 - 3x + 5) = (2)^2 - 3(2) + 5 = 4 - 6 + 5 = 3$
Rational functions: If $r(x) = \frac{p(x)}{q(x)}$ is a rational function, and $q(c) \neq 0$ , then $\lim_{x \to c} r(x) = r(c) = \frac{p(c)}{q(c)}$ .
Example: $\lim_{x \to 1} \frac{x^2 + 1}{x + 3} = \frac{1^2 + 1}{1 + 3} = \frac{2}{4} = \frac{1}{2}$

Special Limits

Limits at Infinity: These describe the behavior of a function as $x$ becomes arbitrarily large (positive or negative): $\lim_{x \to \infty} f(x)$ or $\lim_{x \to -\infty} f(x)$ .
Infinite Limits: These occur when the function’s value grows without bound: $\lim_{x \to c} f(x) = \infty$ or $\lim_{x \to c} f(x) = -\infty$ .
The Squeeze Theorem (or Sandwich Theorem): If $g(x) \le f(x) \le h(x)$ for all $x$ near $c$ (except possibly at $c$ ), and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$ , then $\lim_{x \to c} f(x) = L$ .