Question

The limit below represents the derivative of some function f at some number a. State such an f and a.

lim x-->2 2^x-4/x-2

f(x)=_______

a=_______

Answer 1

I recognized the pattern for derivatives by First Principles

If f(x) = 2^x, then the derivative by First Principles at the point (2,4) would be

Limit (2^x - 2^2)/(x-2) as x ---> 2
which is your starting expression

So f(x) = 2^x and a = 2

check:
lim (2^x - 4)/(x-2) as x ---> 2
= ln2(2^x) by L'Hopital's Rule
= 2.7726

if f(x) = 2^x
then f '(x) = (ln2)(2^x)
f '(2) = ln2(4) = 2.7726

Answer 2

btw, you can also use your calculator to find limits

here is what I did for the above,
1.999999
STO M
2
Y^x
RCL M
=
-
4
=
÷
(
RCL M
-
2
)
=
to get 2.7726