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Limits of Functions

Evaluate the following limits using the epsilon - delta definition and the limit theorems:

a) lim {x -> 0} (x^2 + cos x)/(2 - tan x)

b) lim {x -> sqrt(pi)} ((pi - x^2)^(1/3))/(x + pi)

Solution Preview

a.)
Lt(x->0) {x^2 +cos(x)}/{2 - tan(x)}
= Lt(h->0){(0+h)^2 + cos(0+h)}/(2 - tan(0+h))
= Lt(h->0) {h^2 + cos(h)}{2-tan(h)}
= {0 + cos(0)}/{2-tan(0)}
= 1/2
Let, there exists e >0 such that,
0 < |{x^2 +cos(x)}/{2 - tan(x)} - 1/2| < e
and, d>0 such that,
0<|x-0|<d
=> 0<|x|<d

because,
|x| < d
=> x^2 = |x|^2 < d^2
x^2 +cos(x) < d^2 + 1
let for small x,
2 - tan(x) ~ ...

Solution Summary

The epsilon-delta definition and limit theorems are used to evaluate limits.

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