Use the precise definition of infinite limits to prove the following limits.lim⁡x→0(1x2+1)=∞{\displaystyle\lim_{x\to0}}\left(\frac{1}{$$x^2$$}+1\right)=\infty

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problema 2.7.47

Capitolo 2, Problema 2.7.47

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

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

Guida verificata passo dopo passo

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.

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.

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 $.

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}} $.

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.

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Infinite Limits

Infinite limits occur when the value of a function increases without bound as the input approaches a certain point. In this context, we analyze the behavior of the function as x approaches 0. If the function's value grows larger and larger, we denote this behavior as approaching infinity, which is a key aspect of understanding limits in calculus.

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