Finding the value of delta to verify the limit of f(x) [f(x)=x^2 - 7] as it approaches 3. Epsilon equals .05

this is the only bit of your question that I understand 

 

 

$$\\|x^2 - 7| <0.5\\\\
-0.5 -0.5+7 6.5 6.5 x^2>6.5\qquad or\quad x^2<7.5\\\\$$

 

 

$$[x<-\sqrt{6.5}\quad\cup\quad x>\sqrt{6.5}\quad ]\quad \cap \quad[ -\sqrt{7.5}

 

this is easiest to see in a graph

https://www.desmos.com/calculator/dvyweqgogy

 

$$\\-\sqrt{7.5}