Q. 33 - Calculus

Question

For each limit statement , use algebra to find δ > 0 in terms of $\epsilon$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $\epsilon$.

$\underset{x\to 0}{\lim }(5x^2-1)=-1$

Step-by-Step Solution

Answer

$\delta =\sqrt{\frac{\epsilon }{5}}$

Step1. Given information.

We have been given a limit statement as $\underset{x\to 0}{\lim }(5x^2-1)=-1$.

We have to find $\delta \mathrm{in} \mathrm{terms} \mathrm{of} \epsilon$.

Step 2. Use algebra.

From the given limit statement, we can identify 

$$\begin{gathered} f\left(x\right)=5x^2-1 \\ c=0 \\ L=-1 \\ \mathrm{For} \epsilon >0 \\ \left|\left(5x^2-1\right)-(-1)\right|<\epsilon \\ \left|5x^2-1+1\right|<\epsilon \\ \left|5x^2\right|<\epsilon \\ 5\left|x^2\right|<\epsilon \\ \left|x^2\right|<\frac{\epsilon }{5} \\ \left|x\right|<\sqrt{\frac{\epsilon }{5}} \\ For 0<|x-0|<\delta , we get \left|x\right|<\sqrt{\frac{\epsilon }{5}} \\ \mathrm{Therefore}, \delta =\sqrt{\frac{\epsilon }{5}} \end{gathered}$$