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Prove using the precise definition of the limit

  • #1

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
 

Answers and Replies

  • #2
LCKurtz
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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.
 

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