Question

Find the limit. $\lim _{x \rightarrow 3^{+}} \frac{x^{2}}{x^{2}-9}$

   Find the limit.
 $\lim _{x \rightarrow 3^{+}} \frac{x^{2}}{x^{2}-9}$

Calculus

Ron Larson, Robert… 8th Edition

Chapter 1, Problem 35

Instant Answer

Solved by Expert Lucas Finney

09/22/2021

Step 1

However, we can't directly substitute $x=3$ into the function because it would make the denominator zero and the function undefined. So, we need to consider the limit as $x$ approaches $3$ from the right. Show more…

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Find the limit. $\lim _{x \rightarrow 3^{+}} \frac{x^{2}}{x^{2}-9}$

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Key Concepts

Vertical Asymptote

A vertical asymptote occurs when the function approaches infinity or negative infinity as the variable approaches a certain value. In the context of rational functions, if the denominator tends to zero while the numerator remains non-zero, it typically indicates the presence of a vertical asymptote, which guides the behavior of the function near that point.

Infinite Limit

An infinite limit indicates that the values of a function increase or decrease without bound as the variable approaches a certain point. This concept is essential when analyzing the behavior of functions near points where they are not defined, often due to the denominator approaching zero in rational functions.

One-Sided Limit

A one-sided limit examines the behavior of a function as the variable approaches a specific value from one side only—either from the right (denoted as x ? a?) or the left (denoted as x ? a?). This concept is crucial in establishing the behavior of functions at boundary points or at points of discontinuity, where the values of the function might differ depending on the direction of approach.

Rational Function Limits

Evaluating the limits of rational functions involves examining the behavior of both the numerator and the denominator as the variable approaches a particular value. Special attention is given when the denominator approaches zero, which can lead to unbounded behavior or the emergence of discontinuities.

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