Delta-epsilon proof for limit of 2^(1/(x-1)) as x→1+

[Bigo75]

Homework Statement

A) lim as x-1 of x^2-x+1/x+1 =1/2

B)lim as x-0 of x^2 sin(1/x)=0

C) lim as x-1+ of 2^1/x-1=infinity

Homework Equations

By delta-epsilon proofs

The Attempt at a Solution

A) I took |f(x)-l| and could not factor out an x-1

B) I know that x^2 goes to 0 and that the compositions of functions have the same limit.

C) I have no clue how to evaluate limits from the right hand side!

 

[ircdan]

for B, |x^2sin(1/x)| = |x^2||sin(1/x)| <= ...


what do you know about the sine function? for all u we have |sin(u)| <= ...?

 

[Bigo75]

that it is bounded by 1

 

[ircdan]

right, so let e > 0, and choose d = sqrt(e), then if |x| < d, |x^2sin(1/x)| <= |x|^2*1 < d^2 = e, i'll let you do the rest, but that's the idea though

 

[Math_Geek]

C) looks really hard, I wish I could help you.