find limit of h->0 [f(8+h)-f(8)]/(h) where f(x)=sqrt(x)-8
f(8+h) = sqrt (8+h) - 8
f(8) = sqrt(8) - 8
subtract
sqrt (8+h) -sqrt (8)
but
d/dh (8+h)^.5 = .5 (8+h)^-.5
so
sqrt (8+h) = sqrt 8 + .5 h/sqrt(8+h)
subtract sqrt 8
.5 h/sqrt(8+h)
divide by h
.5/sqrt (8+h)
let h -->0
.5/sqrt 8
To find the limit of the expression as h approaches 0, let's first compute the value of f(8).
f(x) = √(x) - 8
So, f(8) = √(8) - 8
Calculating √(8), we get:
f(8) = 2√2 - 8
Now, let's compute f(8+h):
f(x) = √(x) - 8
f(8+h) = √(8+h) - 8
We'll substitute these values back into the expression:
[f(8+h) - f(8)] / h = [√(8+h) - 8 - (2√2 - 8)] / h
Next, simplify the numerator:
= (√(8+h) - 2√2) / h
Now, factor out a common square root of 2:
= (√(8+h) - √8*√(h)) / h
Notice that √8 simplifies to 2√2:
= (√(8+h) - 2√2 * √h) / h
Next, multiply the numerator and denominator by the conjugate of the numerator:
= [(√(8+h) - 2√2 * √h) * (√(8+h) + 2√2 * √h)] / h * (√(8+h) + 2√2 * √h)
This will allow us to eliminate the square root from the numerator:
= [(8 + h) - 4h] / h * (√(8+h) + 2√2 * √h)
Now, simplify the numerator:
= (8 + h - 4h) / h * (√(8+h) + 2√2 * √h)
= (8 + h - 4h) / h * (√(8+h) + 2√2 * √h)
= (8 - 3h) / h * (√(8+h) + 2√2 * √h)
Finally, take the limit as h approaches 0:
lim(h->0) [(8 - 3h) / h * (√(8+h) + 2√2 * √h)]
Let's evaluate this limit:
lim(h->0) [(8 - 3h) / h * (√(8+h) + 2√2 * √h)]
= [(8 - 3(0)) / (0)] * (√(8+0) + 2√2 * √0)
= 8 * (√8 + 2√2 * 0)
= 8 * (√8 + 0)
= 8 * √8
Therefore, the limit of the expression as h approaches 0 is 8√8.