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Chapter 2: Problem 31

Use the precise definition of infinite limits to prove the following limits. $$\lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}+1\right)=\infty$$

Short Answer

Expert verified
Question: Using the precise definition of infinite limits, prove that $$\lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}+1\right) = \infty$$ Answer: Choose 未 such that $$0 < |x| < \sqrt{\frac{1}{M - 1}}$$ where M is an arbitrarily large positive number.

Step by step solution

01

Rewrite the inequality using M

We want to find 未 such that: $$\frac{1}{x^{2}}+1 > M$$ Let's subtract 1 from both sides of the inequality: $$\frac{1}{x^{2}} > M - 1$$
02

Find a relationship between 未 and M

We want to find a 未 that depends on M. To do this, we first rewrite the inequality as: $$x^{2} < \frac{1}{M - 1}$$ Now we can say that if we choose 未 such that: $$0 < |x| < \sqrt{\frac{1}{M - 1}}$$ Then we have: $$\frac{1}{x^{2}} > M - 1$$
03

Proving the limit using the precise definition

We have found a relationship between 未 and M as follows: $$\delta = \sqrt{\frac{1}{M - 1}}$$ For any arbitrarily large positive number M, if we choose 未 according to the above relationship, then we have: $$0 < |x| < \delta \Rightarrow \frac{1}{x^{2}} > M - 1$$ Hence: $$\frac{1}{x^{2}} + 1 > M$$ Therefore, we have proven using the precise definition of infinite limits that $$\lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}+1\right) = \infty$$

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