One-sided limits are crucial in understanding the behavior of functions as they approach a specific point from either the left or the right side. This concept is foundational for analyzing functions graphically and analytically, especially when a two-sided limit does not exist.

What are One-Sided Limits?

One-sided limits examine how a function's values behave as the input approaches a particular point from one side only:

• Left-hand limit (LHL): $\lim_{x \to c^-} f(x)$ considers values of $x$ approaching $c$ from the left.
• Right-hand limit (RHL): $\lim_{x \to c^+} f(x)$ considers values of $x$ approaching $c$ from the right.

These limits are fundamental for identifying discontinuities and understanding function behavior near specific points.

Calculating One-Sided Limits

Example 1:

Consider the function $f(x) = \frac{x^2 - 4}{x - 2}$.

Find the left-hand limit as $x$ approaches 2, $\lim_{x \to 2^-} f(x)$.

$\begin{aligned} f(x) &= \frac{x^2 - 4}{x - 2} \\ &= \frac{(x + 2)(x - 2)}{x - 2} \quad \text{(Factor the numerator)} \\ &= x + 2 \quad \text{(Simplify, for \(x \neq 2\))} \\ \lim_{x \to 2^-} f(x) &= \lim_{x \to 2^-} (x + 2) \\ &= 2 + 2 \\ &= 4 \end{aligned}$<p></p><h4><strong>Find the right-hand limit as </strong>$x$<strong> approaches 2, </strong>$\lim_{x \to 2^+} f(x)$<strong>.</strong></h4><p></p>$\begin{aligned} \lim{x \to 2^+} f(x) &= \lim{x \to 2^+} (x + 2) \\ &= 2 + 2 \\ &= 4 \end{aligned} $<p></p><h3><strong>Example 2: Handling Infinite Limits</strong></h3><p>Consider$g(x) = \dfrac{1}{(x - 1)^2}$.</p><p></p><h4><strong>Find </strong>$\lim{x \to 1^+} g(x)$<strong><em> </em>and<em> </em></strong>$\lim{x \to 1^-} g(x)$<strong>.</strong></h4><p></p>$\begin{aligned} \lim{x \to 1^+} g(x) &= \lim{x \to 1^+} \frac{1}{(x - 1)^2} \\ &= +\infty \quad \text{(As (x) approaches 1 from the right, the denominator approaches 0, making the fraction infinitely large.)} \\ \lim{x \to 1^-} g(x) &= \lim{x \to 1^-} \frac{1}{(x - 1)^2} \\ &= +\infty \quad \text{(Similarly, as (x) approaches 1 from the left.)} \end{aligned}$<h2 id="recognizing-when-one-sided-limits-differ"><strong>Recognizing When One-Sided Limits Differ</strong></h2><h3><strong>Example 3: Discontinuous Function</strong></h3><p>Let$h(x) = \begin{cases} x^2 & \text{for } x < 3 \\ 7 & \text{for } x \geq 3 \end{cases}$.</p><p></p><h4><strong>Evaluate </strong>$\lim{x \to 3^-} h(x)$<strong><em> </em>and </strong>$\lim{x \to 3^+} h(x)$<strong>.</strong></h4><p></p>$\begin{aligned} \lim_{x \to 3^-} h(x) &= \lim_{x \to 3^-} x^2 \\ &= 3^2 \\ &= 9 \\ \lim_{x \to 3^+} h(x) &= 7 \quad \text{(Since for \(x \geq 3\), \(h(x) = 7\).)} \end{aligned}$<p></p><h2 id="practice-questions"><strong>Practice Questions</strong></h2><h3><strong>Question 1</strong></h3><p><strong>Given</strong>$f(x) = \dfrac{3x - 6}{x - 2}$, <strong>find</strong>$\lim{x \to 2^-} f(x)$<em> </em><strong>and</strong><em> </em>$\lim{x \to 2^+} f(x)$.</p><p></p><h3><strong>Question 2</strong></h3><p><strong>For</strong>$g(x) = \sqrt{x - 4}$, <strong>evaluate</strong>$\lim_{x \to 4^+} g(x)$.</p><p></p><h3><strong>Question 3</strong></h3><p><strong>If</strong>$h(x) = \begin{cases} x^2 & \text{for } x < 1 \\ 3 - x & \text{for } x \geq 1 \end{cases}$, <strong>determine</strong> (\lim<em>{x \to 1^-} h(x)) <strong>and</strong> (\lim</em>{x \to 1^+} h(x)).</p><p></p><h2 id="solutions-to-practice-questions"><strong>Solutions to Practice Questions</strong></h2><h3><strong>Solution to Question 1</strong></h3><h4><strong>Find </strong>$\lim_{x \to 2^-} f(x)$<strong>.</strong></h4><p></p>$\begin{aligned} f(x) &= \frac{3x - 6}{x - 2} \\ &= \frac{3(x - 2)}{x - 2} \quad \text{(Factor out the 3 from the numerator)} \\ &= 3 \quad \text{(for \(x \neq 2\))} \\ \lim_{x \to 2^-} f(x) &= 3 \end{aligned}$<p></p><h4><strong>Find </strong>$\lim_{x \to 2^+} f(x)$.</h4><p>$\begin{aligned} \lim_{x \to 2^+} f(x) &= 3 \end{aligned} $</p><p></p><h3><strong>Solution to Question 2</strong></h3><h4><strong>Evaluate </strong>$\lim_{x \to 4^+} g(x)$<strong>.</strong></h4><p></p>$\begin{aligned} g(x) &= \sqrt{x - 4} \\ \lim_{x \to 4^+} g(x) &= \sqrt{4 - 4} \\ &= \sqrt{0} \\ &= 0 \end{aligned}$<p> </p><h3><strong>Solution to Question 3</strong></h3><h4><strong>Determine </strong>$\lim_{x \to 1^-} h(x)$.</h4><p></p>$\begin{aligned} h(x) &= x^2 \quad \text{(for \(x < 1\))} \\ \lim_{x \to 1^-} h(x) &= (1)^2 \\ &= 1 \end{aligned}$<p></p><h4><strong>Determine </strong>$\lim_{x \to 1^+} h(x)$.</h4><p></p>$\begin{aligned} h(x) &= 3 - x \quad \text{(for \(x \geq 1\))} \\ \lim_{x \to 1^+} h(x) &= 3 - 1 \\ &= 2 \end{aligned}$