Question

Find the limit. $\lim _{x \rightarrow-3^{-}} \frac{x^{2}+2 x-3}{x^{2}+x-6}$

   Find the limit.
$\lim _{x \rightarrow-3^{-}} \frac{x^{2}+2 x-3}{x^{2}+x-6}$

Calculus

Ron Larson, Robert… 8th Edition

Chapter 1, Problem 37

Instant Answer

Solved by Expert Lucas Finney

09/22/2021

Step 1

The numerator $x^{2}+2x-3$ can be factored as $(x-1)(x+3)$ and the denominator $x^{2}+x-6$ can be factored as $(x+3)(x-2)$. So, the expression becomes: \[ \frac{x^{2}+2x-3}{x^{2}+x-6} = \frac{(x-1)(x+3)}{(x+3)(x-2)} \] Show more…

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Find the limit. $\lim _{x \rightarrow-3^{-}} \frac{x^{2}+2 x-3}{x^{2}+x-6}$

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Key Concepts

One-Sided Limits

One-sided limits study the behavior of a function as the input approaches a particular value from one side only (either from the left or right). This concept is crucial when the function behaves differently on each side of the point of interest, often due to discontinuities or asymptotes.

Rational Functions

Rational functions are ratios of polynomials. Understanding their limits involves analyzing the behavior of both the numerator and denominator independently, particularly near points where the denominator approaches zero, which can lead to indeterminate forms or vertical asymptotes.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. In the context of limits, factorization can help simplify the expression by canceling common factors, potentially resolving indeterminate forms and clarifying the function's behavior near the limit point.

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator equals zero while the numerator does not, indicating that the function's value increases or decreases without bound as it approaches the critical point. Understanding the behavior near vertical asymptotes is essential for evaluating limits in such cases.

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