Question

Because of their connection with secant lines, tangentlines, and instantaneous rates, limits of the form

lim f(x + h) - f(x)h occur frequently in calculus. Evaluatet his limit for the given value of x and function f.
h→0

f(x)= 3√ x +5, x= 16

a. Does not exist
b. 24
c. 3/8
d.6

Answer 1

C. 3/8

Explanation:

Let
`f'` be defined as:

`f'= \lim_(h \to 0) (f(x+h)-f(x))/(h)`

Where:

`f(x) = 3\cdot √(x)+5` and
`f(x+h) =3\cdot √(x+h)+5`

The definition is now expanded:

`f' = \lim_(h \to 0) (3\cdot √(x+h)-3\cdot √(x))/(h)`

By rationalization:

`f' = 3\cdot \lim_(h \to 0) (( √(x+h)-√(x))\cdot (√(x+h)+√(x)))/(h\cdot (√(x+h)+√(x)))`

`f' = 3\cdot \lim_(h \to 0) (h)/(h\cdot (√(x+h)+√(x)))`

`f' = 3\cdot \lim_(h \to 0) (1)/(√(x+h)+√(x))`

`f' = 3\cdot \lim_(h \to 0) (1)/((x+h)^(1/2)+x^(1/2))`

If
`h = 0`, then:

`f' = (3)/(2\cdot x^(1/2))`

`f' = (3)/(2\cdot √(x))`

Let evaluate
`f'`when
`x = 16`:

`f'(16) = (3)/(2\cdot √(16))`

`f'(16) = (3)/(8)`

Which corresponds to option C.