Q. 81 - Calculus

Question

Prove that for all x within 0.01 of the value x = 1, the quantity $\frac{1}{{\left(x-1\right)}^2}$ is greater than 10, 000. What does this have to do with

$\underset{x\to 1}{\lim }\frac{1}{{\left(x-1\right)}^2}?$

Step-by-Step Solution

Answer

The given statement is proved. The value of the limit $\underset{x\to 1}{\lim }\frac{1}{{\left(x-1\right)}^2}=\infty .$

Step 1. Given Information.

The given quantity is $\frac{1}{{\left(x-1\right)}^2}.$

Step 2. Prove.

Let the function is $f\left(x\right)=\frac{1}{{\left(x-1\right)}^2}.$

Take the limit of the above function as $x\to 0.01:$

$$\begin{gathered} \underset{x\to 0.01}{\lim }f\left(x\right)=\underset{x\to 0.01}{\lim }\frac{1}{{\left(x-1\right)}^2} \\ \underset{x\to 0.01}{\lim }f\left(x\right)=\frac{1}{{\left(0.01-1\right)}^2} \\ \underset{x\to 0.01}{\lim }f\left(x\right)=1.02 \end{gathered}$$

Now, take the limit of the above function as $x\to 1:$

$$\begin{gathered} \underset{x\to 1}{\lim }f\left(x\right)=\underset{x\to 1}{\lim }\frac{1}{{\left(x-1\right)}^2} \\ \underset{x\to 1}{\lim }f\left(x\right)=\frac{1}{{\left(1-1\right)}^2} \\ \underset{x\to 1}{\lim }f\left(x\right)=\frac{1}{0} \\ \underset{x\to 1}{\lim }f\left(x\right)=\infty \end{gathered}$$

Therefore, the quantity $\frac{1}{{\left(x-1\right)}^2}$ is greater than $10,000.$

Step 3. Finding the limit.

Let's find the limit:

$$\begin{gathered} \underset{x\to 1}{\lim }f\left(x\right)=\underset{x\to 1}{\lim }\frac{1}{{\left(x-1\right)}^2} \\ \underset{x\to 1}{\lim }f\left(x\right)=\frac{1}{{\left(1-1\right)}^2} \\ \underset{x\to 1}{\lim }f\left(x\right)=\frac{1}{0} \\ \underset{x\to 1}{\lim }f\left(x\right)=\infty \end{gathered}$$