Question

Given $f(x)=x^{2}-\frac{3}{x+2},$ find $$\lim _{x \rightarrow-2} f(x)$$

   Given $f(x)=x^{2}-\frac{3}{x+2},$ find $$\lim _{x \rightarrow-2} f(x)$$

Precalculus

John W. Coburn 2nd Edition

Chapter 11, Problem 61

Instant Answer

Solved by Expert Elizabeth Xu

06/08/2020

Step 1

We can do this by substituting values that are slightly less than $-2$ into the function $f(x)$ and observing the trend. For example, let's use $-2.1$, $-2.01$, and $-2.001$. Substituting $x = -2.1$ into $f(x)$, we get $f(-2.1) = (-2.1)^2 - \frac{3}{-2.1 + 2} = Show more…

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Given $f(x)=x^{2}-\frac{3}{x+2},$ find $$\lim _{x \rightarrow-2} f(x)$$

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Key Concepts

Discontinuities

Discontinuities are points where a function is not continuous, meaning there is a break, jump, or other irregularity in its graph. Identifying the type of discontinuity, such as a removable or non-removable discontinuity, is important when evaluating limits, as the behavior of the function near those points can vary significantly.

Limit of a Function

A limit describes the behavior of a function as its input approaches a particular value. It allows us to understand the trend of the function even if the function might not be defined at that exact point. This concept is fundamental in calculus and analysis for studying continuity, derivatives, and the behavior of functions around points of interest.

One-Sided Limits

One-sided limits consider the behavior of a function as the variable approaches a specific value from either the left (approaching from smaller values) or the right (approaching from larger values). They are crucial in situations where the function behaves differently on either side of a point, thus determining whether a two-sided limit exists.

Vertical Asymptotes

Vertical asymptotes occur when a function grows without bound as the variable approaches a certain value usually due to the denominator of a rational function approaching zero. Recognizing vertical asymptotes is critical since they indicate points where the function is not defined and where the limit may diverge to infinity or negative infinity.

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