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    Limits of Functions

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    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)

    © BrainMass Inc. brainmass.com February 24, 2021, 2:14 pm ad1c9bdddf
    https://brainmass.com/math/real-analysis/limits-functions-epsilon-8950

    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. Epsilon is used for a delta definition. The limit theorems are analyzed.

    $2.19

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