The limit as x approaches 0 gets closer to 2 to the x power - 1 over x equals?

Question

Lim 2^x-1/x =

x-->0

A) 0

B) ln2

C) 1

D) 1/ln2

Side Note- the limit as x approaches 0 gets closer to 2 to the x power - 1 over x equals?

Answer 1

Let f(x) = 2^x

Then the given limit can be rewritten as lim_x→0 [ (f(x) - f(0)) / (x - 0)] = f'(0) = ln2(2⁰) = ln2

Answer 2

Use L'Hopitals Rule

lim x>0 = lim x > 0 d/dx (2^x - 1) / d/dx (x)

d/dx a^x = a^x ln a

So the derivative quotient is 2^x ln 2 / 1, which tends to ln 2 as x > 0

So B

You can check this by substituting x = 0.0000001 into the original limit, you should get an answer close to ln 2