Question

Calculus limit problem

Answer 1

Is this the question\[\lim_{h\rightarrow0}\frac{8(\frac{1}{2}+h)^8-8(\frac{1}{2})^8}{h}\]

Answer 2

Yeah

Answer 3

Take out 8 as the common factor and put it outside the limit:
\[8\lim_{h\rightarrow0}\frac{(\frac{1}{2}+h)^8-(\frac{1}{2})^8}{h}\]
Now, use \(a^2 - b^2 = (a+b)(a-b)\) to factorize the numerator:
\[(\frac{1}{2}+h)^8-(\frac{1}{2})^8\]\[=[(\frac{1}{2}+h)^4]^2-[(\frac{1}{2})^4]^2\]\[=[(\frac{1}{2}+h)^4-(\frac{1}{2})^4][(\frac{1}{2}+h)^4+(\frac{1}{2})^4]\]
Use the same identity to factorize \([(\frac{1}{2}+h)^4-(\frac{1}{2})^4]\) again, can you try?

Answer 4

yesss! How do you even see tht? i've been trying to solve that problem for at least an hour! Thank you so much, u just made my life a hell of a lot easier

Answer 5

Experience :)

Answer 6

You're welcome :)