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Chapter 2: Problem 9

Evaluate the following limits. $$\lim _{x \rightarrow \infty}\left(3+\frac{10}{x^{2}}\right)$$

Short Answer

Expert verified
Answer: The limit of the function as x approaches infinity is 3.

Step by step solution

01

Analyze the function

We have a function: $$f(x) = 3 + \frac{10}{x^2}$$ Which consists of two terms: a constant (3) and a term that depends on x: the rational function \(\frac{10}{x^2}\).
02

Determine the behavior of the rational term

As x approaches infinity, the denominator (\({x^2}\)) becomes larger and larger. Therefore, we have: $$\lim_{x \rightarrow \infty} \frac{10}{x^2} = 0$$ This means that the rational term tends to zero as x approaches infinity.
03

Determine the limit of the function

Knowing the behavior of the rational term, we can now determine the limit of the function: $$\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} \left(3 + \frac{10}{x^2}\right)$$ As x approaches infinity, the rational term becomes smaller, tending to zero. Therefore, the function behaves like the constant term (3): $$\lim_{x \rightarrow \infty} f(x) = 3 + \lim_{x \rightarrow \infty} \frac{10}{x^2} = 3 + 0 = 3$$ The limit of the given function as x approaches infinity is 3.

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Most popular questions from this chapter

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