Question

I am not sure how to slove

Answer 1

First .... the limit of the given function is NOT EQUAL to 5 (it diverges so it has no limit). There is a typo. The 14 should be positive.

`\lim_(x \to\ 7) \bigg((x^2-9x+14)/(x-7)\bigg)=5`

The precise definition of a limit is:

`\text{If for every } \epsilon>0\text{ there exists a }\delta >0\text{ such that}\\|f(x)-L|<\epsilon\ \text{whenever }|x-a|<\delta`

Given:

`f(x) = (x^2-9x+14)/(x-7)\\\\L=5\\\\a=7\\\\\\|f(x)-L|<\epsilon\qquad \qquad \text{whenever}\quad |x-a|<\delta\\\\\bigg|(x^2-9x+14)/(x-7)-5\bigg|<\epsilon\qquad \text{whenever}\quad |x-7|<\delta\\\\\\\bigg|((x-2)(x-7))/(x-7)-5\bigg|<\epsilon\qquad \text{whenever}\quad |x-7|<\delta\\\\\\|x-2-5|<\epsilon\qquad \qquad \quad \text{whenever}\quad |x-7|<\delta\\\\|x-7|<\epsilon\qquad \qquad \text{whenever}\quad |x-7|<\delta`

⇒
`\epsilon = \delta`

When ε = 0.1, δ = 0.1

When ε = 0.01, δ = 0.01