Prove using the precise definition of the limit

[TheRedDevil18]

Homework Statement

lim x→2 (x²+1) = 5

Homework Equations

0 < |x-a| < delta

|f(x) - L| < ε

The Attempt at a Solution

0 < |x-2| < delta

|x²-4| < ε

|(x-2) (x+2)| < ε

|x-2| |x+2| < ε.....and I am stuck here, any help

 

[LCKurtz]

Homework Statement

lim x→2 (x²+1) = 5

Homework Equations

0 < |x-a| < delta

|f(x) - L| < ε

The Attempt at a Solution

0 < |x-2| < delta

|x²-4| < ε

|(x-2) (x+2)| < ε

|x-2| |x+2| < ε.....and I am stuck here, any help

Continuing your analysis:$$
|x-2|<\frac \epsilon {|x+2|}$$You are trying to get the right side small by getting $x$ close to $2$. So you don't want the denominator to get close to $0$. Can you keep it away from $0$ if $x$ is close to $2$? That's the idea you need to pursue.