Question

Limits of Rational FunctionsIn Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.f(x) = (x + 1)/(x² + 3)

Accepted Answer

Step 1: To find the limit of the rational function $$ f(x) = \frac{x + 1}{x^2 + 3}  as$$ $$ x 	o \infty $$, start by analyzing the degrees of the numerator and the denominator. The numerator $$ x + 1  is of $$degree 1, and the denominator $$ x^2 + 3  is of $$degree 2. Step 2: Since the degree of the denominator is greater than the degree of the numerator, the limit as $$ x 	o \infty $$ will be determined by the leading terms. Divide each term in the numerator and the denominator by $$ x^2 $$, the highest power of $$ x  in $$the denominator. Step 3: After dividing, the expression becomes $$ \frac{\frac{x}{x^2} + \frac{1}{x^2}}{\frac{x^2}{x^2} + \frac{3}{x^2}} = \frac{\frac{1}{x} + \frac{1}{x^2}}{1 + \frac{3}{x^2}} . As$$ $$ x 	o \infty $$, both $$ \frac{1}{x} $$ and $$ \frac{1}{x^2} $$ approach 0. Step 4: Therefore, the limit of the function as $$ x 	o \infty  is$$ $$ \frac{0 + 0}{1 + 0} = 0 . $$Step 5: To find the limit as $$ x 	o -\infty $$, repeat the same process. The behavior of the function is similar because the terms $$ \frac{1}{x} $$ and $$ \frac{1}{x^2} $$ also approach 0 as $$ x 	o -\infty . $$Thus, the limit is $$ \frac{0 + 0}{1 + 0} = 0 .$$

Limits of Rational Functions

In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.

f(x) = (x + 1)/(x² + 3)

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Key Concepts

Limits

Rational Functions

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