what is the limit of this problem?

Question

Lim h-->0 (((.5 + h)⁵) - ((.5)⁵))/h

Answer 1

This is the derivative of x^5 at x = 0.5.

Lim h-->0 (((.5 + h)⁵) - ((.5)⁵))/h

= 5x^4 at x = 1/2

= 5/16 <==Answer

Answer 2

This problem is given before the students know about derivatives, so it must be solved with algebra.

To find (.5 + h)^5 either multiply four times, or use the binomial theorem with the help of Pascal's Triangle:

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1

(h + 1/2)^5 = h^5 + 5h^4(1/2) + 10h^3(1/2)^2 + 10h^2(1/2)^3 + 5h(1/2)^4 + (1/2)^5

Then (((.5 + h)^5) - ((.5)^5))/h
= (h^5 + 5h^4(1/2) + 10h^3(1/2)^2 + 10h^2(1/2)^3 + 5h(1/2)^4 + (1/2)^5 - (1/2)^5)/h
= h^4 + 5h^3(1/2) + 10h^2(1/2)^2 + 10h(1/2)^3 + 5(1/2)^4

Now as h —> 0 all of the terms with h in them go away and:

(((.5 + h)^5) - ((.5)^5))/h —> 5(1/2)^4 = 5/16.