Mastering Left-Hand and Right-Hand Limits: A Step-by-Step Guide with Examples
Understanding limits is a fundamental concept in calculus. Evaluating left-hand limits (\( \lim_{x \to c^-} \)) and right-hand limits (\( \lim_{x \to c^+} \)) is essential to determine if a limit exists at a given point. This guide walks you through the steps to evaluate these limits and explains when they are equal or unequal, supported by examples to solidify your understanding.
How to Evaluate Left and Right Limits: Step-by-Step Guide
Step 1: Understand the Function and Point of Interest (\(c\))
Identify the function \(f(x)\) and the value \(c\) at which you want to evaluate the limit. Determine if \(f(x)\) is continuous, piecewise-defined, or has discontinuities near \(c\).
Step 2: Find the Left-Hand Limit (\( \lim_{x \to c^-} f(x) \))
Approach the value of \(c\) from the left (values less than \(c\)). Substitute \(x\)-values slightly less than \(c\) into \(f(x)\) and observe the behavior.
Step 3: Find the Right-Hand Limit (\( \lim_{x \to c^+} f(x) \))
Approach \(c\) from the right (values greater than \(c\)). Substitute \(x\)-values slightly greater than \(c\) into \(f(x)\) and observe the behavior.
Step 4: Compare the Two Limits
If \( \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) \), the limit exists, and \(\lim_{x \to c} f(x)\) equals this common value. If \( \lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x) \), the limit does not exist.
When Do Left and Right Limits Differ?
Left and Right Limits Are Equal When:
- The Function Is Continuous at \(c\): Functions without breaks or jumps, such as polynomials or trigonometric functions.
Example: \( \lim_{x \to 3^-} (x + 2) = \lim_{x \to 3^+} (x + 2) = 5 \).
- Piecewise Functions Have Consistent Values: If a piecewise-defined function is consistent at the boundary.
Example: \( f(x) = \begin{cases} x + 1, & \text{if } x \leq 1, \\ x + 1, & \text{if } x > 1 \end{cases} \) at \(x = 1\), both left and right limits are 2.
- Symmetry in Function Behavior: Symmetrical functions like \(f(x) = |x|\) show equal left and right limits.
Example: \( \lim_{x \to 0^-} \frac{1}{x^2} = \lim_{x \to 0^+} \frac{1}{x^2} = \infty \).
Left and Right Limits Differ When:
- Jump Discontinuities: A piecewise function has different definitions on either side of a boundary.
Example: \( f(x) = \begin{cases} x^2, & \text{if } x < 1, \\ 2x + 1, & \text{if } x \geq 1 \end{cases} \). At \(x = 1\), \( \lim_{x \to 1^-} f(x) = 1 \) and \( \lim_{x \to 1^+} f(x) = 3 \).
- Infinite Discontinuities: The function approaches \(-\infty\) from one side and \(+\infty\) from the other.
Example: \( f(x) = \frac{1}{x} \). At \(x = 0\), \( \lim_{x \to 0^-} f(x) = -\infty \) and \( \lim_{x \to 0^+} f(x) = +\infty \).
Examples for Practice
Example 1: Limit Exists
Evaluate \(\lim_{x \to 2} f(x)\), where \(f(x) = x^2 - 3x + 4\).
Solution:
\(\lim_{x \to 2^-} f(x) = 2^2 - 3(2) + 4 = 2 \).
\(\lim_{x \to 2^+} f(x) = 2^2 - 3(2) + 4 = 2 \).
Both limits are equal, so \(\lim_{x \to 2} f(x) = 2\).
Example 2: Limit Does Not Exist
Evaluate \(\lim_{x \to 0} f(x)\), where \(f(x) = \frac{|x|}{x}\).
Solution:
\(\lim_{x \to 0^-} f(x) = \frac{|x|}{x} = -1, \quad \lim_{x \to 0^+} f(x) = \frac{|x|}{x} = 1 \).
The left and right limits differ, so the limit does not exist.