Chapter 2.7, Problem 3QC

In Example 7, if N is increased by a factor of 100, how must δ change?

Example 7 An Infinite Limit Proof

Let $f(x)=\frac{1}{{(x-2)}^2}$. Prove that $\underset{x\to 2}{\lim }f(x)=\infty$.

Solution

Step 1:

Find δ > 0. Assuming N > 0, we use the inequality $\frac{1}{{(x-2)}^2}>N$ to find δ, where δ depends only on N. Taking reciprocals of this inequality, it follows that

${(x-2)}^2<\frac{1}{N}$

$\left|x-2\right|<\frac{1}{\sqrt{N}}.$    Take the square root of both sides.

The inequality $\left|x-2\right|<\frac{1}{\sqrt{N}}.$ has the form |x − 2| < δ if we let $\delta <\frac{1}{\sqrt{N}}$. We now write a proof based on this relationship between δ and N.

Step 2:

Write a proof. Suppose N > 0 is given. Let $\delta =\frac{1}{\sqrt{N}}$ and assume $0<\left|x-2\right|<\delta =\frac{1}{\sqrt{N}}$. Squaring both sides of the inequality $\left|x-2\right|<\frac{1}{\sqrt{N}}$ and taking reciprocals, we have

${(x-2)}^2<\frac{1}{N}$    Square both sides.

$\frac{1}{{(x-2)}^2}>N.$    Take reciprocals of both sides.

We see that for any positive N, if $0<\left|x-2\right|<\delta =\frac{1}{\sqrt{N}}$, then $f(x)=\frac{1}{{(x-2)}^2}>N$. It follows that $\underset{x\to 2}{\lim }\frac{1}{{(x-2)}^2}=\infty$. Note that because $\delta =\frac{1}{\sqrt{N}}$, δ decreases as N increases.