Question

If f is a function such that lim (f(x)-f(a))/(x-a)=7 as x->a, then which of the following statements must be true?

Answer 1

f(a) = 7
f(x) is continuous at x = 7
the linear approximation for f(x) at x = a is 7
f '(a) = 7

Answer 2

Here is one of the limit definitions for the derivative.\[\large\rm f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\]

Answer 3

So if we have this:\[\large\rm f'(a)=\color{orangered}{\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=7}\]What does that tell you?

Answer 4

no we arent

Answer 5

Strange.. because it's clearly the last option.
But the second also seems to be true.

Differentiability implies continuity.
So our function should be continuous at x=7 since the derivative exists there.

Answer 6

hm ok

Answer 7

"Differentiability implies continuity.
So our function should be continuous at x=7 since the derivative exists there."

"f(x) is continuous at x = 7"



Might not be true. What we do know is that \(f(x)\) is continuous at \(x=a\)


therefore the only answer must be \(f '(a) = 7\)

Answer 8

Oh continuous at x=a, yes yes yes c:
Another derp moment. Zark to save the day.