Prove that the limit . . .?

Question

Proving limits really isn't my strong point . . . so, any help would be appreciated.

Problem:
Prove, using Definition 6, that the limit as x --> -3 of 1 / (x + 3)^4 equals ∞.

(I'm sure it isn't totally necessary to those of you who know what you're doing - but Definition 6 in our book says: Let f be a function defined on some open interval that contains the number a, except possibly a itself. Then: the limit as x --> a of f(x) = ∞ means that for every positive number M, there is a positive number δ such that if 0< |x - a| <δ then f(x) > M)

Ron W · Accepted Answer

Let M be given.

Suppose 0 < |x - (-3)| < δ Then
1/|x + 3| > 1/δ
(1/|x + 3|)^4 > 1/δ^4

Because we're taking expressions to an even power, we can drop the absolute values.

(1/(x + 3))^4 > 1/δ^4 = M

Thus, if we choose δ = 1/M^¼ (or less), (x - (-3)) < δ → (1/(x + 3))^4 > M

Prove that the limit . . .?

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