Home > Calculus > Determining Limits Algebraically

What's the limit of #(x^n-a^n)/(x-a)# as #x# approaches #a# , using the derivate number which is #f'(a)# ?

Answer 1

Paisley Johnson

Please see below.

I'm not sure I understand your question. I don't know what you mean by "using the derivate number which is #f'(a)#"

If I understand it, I think I need to point out that

#lim_(xrarra)(x^n-a^n)/(x-a) = f'(a)# for #f(x) = x^n#.

That is:

For #f(x) = x^n#,

#f'(a) = lim_(xrarra)(f(x) - f(a))/(x-a) = (x^n-a^n)/(x-a)#

We also know, by the power rule for derivatives,

That for #f(x) = x^n#, we haved #f'(x) = nx^(n-1)# and #f'(a) = na^(n-1)#.

We can conclude that

#lim_(xrarra)(x^n-a^n)/(x-a) = na^(n-1)#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Isaiah Allen

The limit of (x^n-a^n)/(x-a) as x approaches a is n*a^(n-1).

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.