How do I evaluate lim as h->0 [5((1/2)+h)^4 - 5(1/2)^4]/h

Answer 1

Also , another similar question that I am having trouble with is lim as x->0 (tan^3 (2x))/(x^3)

Answer 2

binomial expansion of (.5+h)^4
1* .5^4 + 4* .5^3 h + 6* .5^2 h^2 etc

5 [ .5^4 + 4* .5^3 h + 6* .5^2 h^2 + 4 .5 h^3 + h^4 -.5^4 ] / h

only term left when h--> 0 is
5 [ 4*.5^3 ]

Answer 3

by definition
dy/dx = lim (f(x+h) - f(x))/h as h ---> 0

if you look at your expression carefully, you can see that they are attempting to find dy/dx of f(x) = 5x^4 when x=1/2

so dy/dx = 20x^3, which at x= 1/2 = 20/8 = 5/2

just like in Damon's answer.

Answer 4

I figured maybe the derivative is in the next chapter :)

Answer 5

Thanks for the help, I know what to do now. :D