Question

Evaluate $\lim _{x \rightarrow \infty}$ for the function. $$f(x)=\frac{x+3}{2-x}$$

Name: Problem 43, Chapter 1: Calculus Single Variable
Uploaded: 2021-04-28T21:33:26-08:00
Duration: 1 min 5 s
Channel: Prashansha Kaushik
Description: Evaluate limx →∞ for the function. f(x)=(x+3)/(2-x)

   Evaluate $\lim _{x \rightarrow \infty}$ for the function.
$$f(x)=\frac{x+3}{2-x}$$

Calculus Single Variable

Deborah… 7th Edition

Chapter 1, Problem 43

Instant Answer

Solved by Expert Prashansha Kaushik

04/28/2021

Step 1

We take $x$ common in the numerator and the denominator. The function then becomes: $$f(x)=\lim _{x \rightarrow \infty} \frac{x(1+\frac{3}{x})}{x(\frac{2}{x}-1)}$$ Show more…

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Evaluate $\lim _{x \rightarrow \infty}$ for the function. $$f(x)=\frac{x+3}{2-x}$$

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Key Concepts

Dominant Terms

In evaluating limits at infinity for rational functions, the highest degree terms in the numerator and denominator are the dominant factors that influence the function’s behavior. By focusing on these terms, one can simplify the limit calculation, as the lower degree terms become negligible for large values of the variable.

Rational Functions

Rational functions are defined as the quotient of two polynomials. Their long-term behavior, especially as the variable approaches infinity, is largely determined by the degrees and leading coefficients of the numerator and the denominator.

Limits at Infinity

Limits at infinity examine the behavior of functions as the variable grows without bound. This concept helps determine whether a function approaches a particular value, diverges, or oscillates as its input becomes very large.

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