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Chapter 2: Problem 10

Evaluate the following limits. $$\lim _{x \rightarrow \infty}\left(5+\frac{1}{x}+\frac{10}{x^{2}}\right)$$

Short Answer

Expert verified
Answer: The limit of the function (5 + 1/x + 10/x^2) as x approaches infinity is 5.

Step by step solution

01

Identify the term types in the function

Identify the terms in the expression given: constant term (5), a term with x in the denominator (1/x), and a term with x^2 in the denominator (10/x^2).
02

Evaluate the effect of the terms when x approaches infinity

Let's look at each term separately as x approaches infinity: 1. For the 5 (constant term), there will be no change and will remain the same. 2. For the 1/x term, as x approaches infinity, the value of the term approaches 0. 3. For the 10/x^2 term, as x approaches infinity, the value of the term approaches 0.
03

Combine the results to find the limit

Combine the evaluated results of each term as x approaches infinity: $$\lim _{x \rightarrow \infty}\left(5+\frac{1}{x}+\frac{10}{x^{2}}\right)=5+0+0$$
04

Calculate the final answer

Perform the calculations to find the final answer: The limit of the given function as x approaches infinity is: $$\lim _{x \rightarrow \infty}\left(5+\frac{1}{x}+\frac{10}{x^{2}}\right)=5+0+0=5$$

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