# Limit induction proof

I need help to verify the following

Prove that if does not equal 0

lim of x approaches a: x^-n = a^-n

I know to prove lim of x approaches a: x^n = a^n requires induction so I believe that this problem requires the same.

lim of x approaches a: [f(x)]^-n ... = k^-n

inductive step

lim of x approaches a: [f(x)]^-n-1 * lim of x approaches a: [f(x)] =

k^-n-1*k = k

Does this prove the question?

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If the statement is known to you with positive exponents (as you say), then you can prove the statement for negative ones by using that the function $x\mapsto\dfrac1x$ is continuous.