Question

2. The slope of the tangent line to the graph of the exponential function y=2^x at the point (0,1) is the limit of (2^x-1)/x as x approaches 0. Estimate the slope to three decimal places.

My work:


(-1,0.5)
(-0.5,0.586)
(-0.9,0.516)
(-0.999,0.5)
(0,empty)
(0.0001, 0.693)
(0.001, 0.693)
(0.01,0.696)
(1,1)

Using the trend,I averaged the two closest numbers.... I got 0.597..

Answer 1

I am uncertain what you are doing. Estimating is easy on a calculator, put in values of x close to zero as you did on the .0001, .001, and press equals on your calculator. I have no idea why you averaged values values to get .597

Now, later in Calculus, you will learn L'Hopital's rule for forms of 0/0 for values, that rule says the limit of

(2^x-1)/x as x approaches zero is the same limit as
2^x*ln(2)/1 which for x=0, becomes the ln (2)=.693