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=_______
btw, you can also use your calculator to find limits
here is what I did for the above,
1.999999
STO M
2
Yx
RCL M
=
-
4
=
÷
(
RCL M
-
2
)
=
to get 2.7726
To find a function f(x) and a number a such that the given limit represents the derivative of f at a, we need to evaluate the limit using the rules of differentiation.
Taking the limit of the expression as x approaches 2:
lim x->2 (2^x - 4) / (x - 2)
This expression can be simplified further using algebraic manipulation, specifically factoring out the common factor of (2^x - 4) - (x - 2):
lim x->2 [(2^x - 4) - (x - 2)] / (x - 2)
Using the property (a^x - b^x) = (a - b)(a^(x-1) + a^(x-2)b + a^(x-3)b^2 + ... + b^(x-1)):
lim x->2 [2(2^(x-2) - 1)] / (x - 2)
Now, we can see that we have a factor of (x - 2) in both the numerator and the denominator, which can be canceled out:
lim x->2 [2(2^(x - 2) - 1)] / (x - 2) = lim x->2 2(2^(x - 2) - 1) = 2
Therefore, we have f(x) = 2 and a = 2 as a function and number that satisfy the given limit as the derivative of f at a.
To find an f(x) and a that represents the given limit, we can start by simplifying the expression inside the limit.
lim x-->2 (2^x - 4)/(x - 2)
We can see that this limit resembles the derivative of the function 2^x at some number a.
So, let's find the derivative of 2^x and set it equal to the given limit:
f(x) = 2^x
f'(x) = d/dx (2^x)
= (ln 2) * 2^x
Now, we need to find the number a for which the derivative f'(x) is equal to the given limit.
lim x-->2 (2^x - 4)/(x - 2) = f'(a)
To proceed, we need to find a value of a where f'(x) is equal to the given limit lim x-->2 (2^x - 4)/(x - 2).
Let's substitute a value of a into f'(x) and see if it matches the given limit.
f'(2) = (ln 2) * 2^2
= (ln 2) * 4
If we compare this to the given limit, we can see that substituting f(x) = 2^x and a = 2 satisfies the given condition:
lim x-->2 (2^x - 4)/(x - 2) = f'(2) = (ln 2) * 4
Therefore, the function f(x) = 2^x and a = 2 represents the given limit.
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