Q. 84 - Calculus

Question

Use the Squeeze Theorem to find each of the limits in Exercises. Explain exactly how the Squeeze Theorem applies in each case.

$\underset{\mathbf{ }\mathbf{ }\mathbf{x}\mathbf{\to }\mathbf{1}}{\mathbf{ }\mathbf{lim}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathit{c}\mathit{o}\mathit{s}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}$

Step-by-Step Solution

Answer

The limit of the equation $\underset{\mathbf{ }\mathbf{ }\mathbf{x}\mathbf{\to }\mathbf{1}}{\mathbf{ }\mathbf{lim}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathit{c}\mathit{o}\mathit{s}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}$ is 0

Step 1. Given Information:

$\underset{\mathbf{ }\mathbf{ }\mathbf{x}\mathbf{\to }\mathbf{1}}{\mathbf{ }\mathbf{lim}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathit{c}\mathit{o}\mathit{s}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}$

Step 2. Applying Squeeze theorem on the limit:

By applying the Squeeze theorem;

$$\begin{gathered} \mathit{R}\mathit{a}\mathit{n}\mathit{g}\mathit{e}\mathbf{ }\mathit{o}\mathit{f}\mathbf{ }\mathbf{cos}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}\mathbf{ }\mathbf{"}\mathit{o}\mathit{r}\mathbf{ }\mathbf{cos}\mathbf{ }\mathit{x}\mathbf{"}\mathbf{ }\mathit{i}\mathit{s}\mathbf{ }\mathbf{[}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{]} \\ \mathbf{-}\mathbf{1}\mathbf{\le }\mathbf{cos}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}\mathbf{\le }\mathbf{1} \\ \mathit{N}\mathit{o}\mathit{w}\mathbf{ }\mathit{m}\mathit{u}\mathit{l}\mathit{t}\mathit{i}\mathit{p}\mathit{l}\mathit{y}\mathit{i}\mathit{n}\mathit{g}\mathbf{ }{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathbf{ }\mathit{o}\mathit{n}\mathbf{ }\mathit{a}\mathit{l}\mathit{l}\mathbf{ }\mathit{s}\mathit{i}\mathit{d}\mathit{e}\mathbf{;} \\ \mathbf{-}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathbf{\le }{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathbf{cos}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}\mathbf{\le }{\left(x-1\right)}^{\mathbf{2}} \\ \mathbf{-}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{\le }{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathbf{cos}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}\mathbf{\le }{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{1} \\ \mathbf{-}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{x}\mathbf{-}\mathbf{1}\mathbf{\le }{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathbf{cos}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}\mathbf{\le }{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{1} \\ \mathit{A}\mathit{p}\mathit{p}\mathit{l}\mathit{y}\mathit{i}\mathit{n}\mathit{g}\mathbf{ }\mathit{l}\mathit{i}\mathit{m}\mathit{i}\mathit{t}\mathit{s}\mathbf{ }\mathit{o}\mathit{n}\mathbf{ }\mathit{a}\mathit{l}\mathit{l}\mathbf{ }\mathit{s}\mathit{i}\mathit{d}\mathit{e}\mathbf{ }\mathit{a}\mathit{s}\mathbf{ }\mathit{x}\mathbf{\to }\mathbf{1}\mathbf{;} \\ \underset{\mathbf{x}\mathbf{\to }\mathbf{1}}{\mathbf{lim}}\mathbf{-}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{x}\mathbf{-}\mathbf{1}\mathbf{\le }\underset{\mathbf{x}\mathbf{\to }\mathbf{1}}{\mathbf{lim}}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{cos}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}\mathbf{\le }\underset{\mathbf{x}\mathbf{\to }\mathbf{1}}{\mathbf{lim}}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{1} \\ \mathbf{-}\mathbf{1}\mathbf{+}\mathbf{2}\mathbf{-}\mathbf{1}\mathbf{\le }\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{2}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{cos}\frac{\mathbf{1}}{\mathbf{1}\mathbf{-}\mathbf{1}}\mathbf{\le }\mathbf{1}\mathbf{-}\mathbf{2}\mathbf{+}\mathbf{1} \\ \mathbf{0}\mathbf{\le }\mathbf{0}\mathbf{(}\mathbf{cos}\mathbf{ }\mathbf{0}\mathbf{)}\mathbf{\le }\mathbf{0} \\ \mathbf{0}\mathbf{\le }\mathbf{0}\mathbf{\le }\mathbf{0} \\ \mathit{s}\mathit{o}\mathbf{,}\underset{\mathbf{ }\mathbf{ }\mathbf{x}\mathbf{\to }\mathbf{1}}{\mathbf{ }\mathbf{lim}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathbf{cos}\frac{\mathbf{1}}{\mathbf{x}\mathbf{-}\mathbf{1}}\mathbf{=}\mathbf{0} \end{gathered}$$

Step 3. Graph Representation: