Use the precise definition of infinite limits to prove the following limits.


$\underset{x\to 0}{\lim }(\frac{1}{x^2}+1)=\infty$

Verified step by step guidance

1. Step 1: Understand the definition of an infinite limit. The limit of a function f(x) as x approaches a value c is infinity if for every positive number M, there exists a δ > 0 such that if 0 < |x - c| < δ, then f(x) > M.
2. Step 2: Identify the function and the point of interest. Here, the function is f(x) = \( \frac{1}{x^2} + 1 \) and we are interested in the behavior as x approaches 0.
3. Step 3: Set up the inequality based on the definition. We need to show that for every M > 0, there exists a δ > 0 such that if 0 < |x| < δ, then \( \frac{1}{x^2} + 1 > M \).
4. Step 4: Simplify the inequality \( \frac{1}{x^2} + 1 > M \) to \( \frac{1}{x^2} > M - 1 \). This implies \( x^2 < \frac{1}{M - 1} \) and thus |x| < \( \frac{1}{\sqrt{M - 1}} \).
5. Step 5: Choose δ = \( \frac{1}{\sqrt{M - 1}} \). This choice of δ ensures that whenever 0 < |x| < δ, the inequality \( \frac{1}{x^2} + 1 > M \) holds, proving the limit is infinity.