Q. 44 - Calculus

Question

For each limit statement in Exercises $41-44$, use algebra to find $\delta$ or $N$ in terms of $\epsilon$ or $M$, according to the appropriate formal limit definition.

$\underset{x\to \infty }{\lim }\left(x^2+2\right)=\infty$,find $N$ in terms of $M$.

Step-by-Step Solution

Answer

For $\underset{x\to \infty }{\lim }\left(x^2+2\right)=\infty$, $N=\sqrt{M-2}$.

Step 1 . Given information

$\underset{x\to \infty }{\lim }\left(x^2+2\right)=\infty$.

Step 2 . From the limit expression,

$$\begin{gathered} f\left(x\right)=x^2+2 \\ c=\infty \\ L=\infty \end{gathered}$$

Now for $\delta >0$.

$$\begin{gathered} \left|x\right|<N \\ {\left|x\right|}^2<N^2 \\ \left|x^2+2\right|<N^2+2 \\ \left|x^2+2\right|<M \end{gathered}$$

Therefore, 

$$\begin{gathered} N^2+2=M \\ {N}^2=M-2 \\ N=\sqrt{M-2} \end{gathered}$$