Question

If f is a function such that the limit as x approaches a of the quotient of the quantity f of x minus f of a and the quantity x minus a equals 5, then which of the following statements must be true?

f(a) = 5
The slope of the tangent line to the function at x = a is 5.
The slope of the secant line through the function at x = a is 5.
The linear approximation for f(x) at x = a is y = 5

Answer 1

Will medal and fan!

Answer 2

\[\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}=5\] is the given right

Answer 3

I will give you a hint isn't the left hand side the definition of a magical d word.

Answer 4

The answer is the second one, I believe.

Answer 5

yeah \[\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}=f'(a)\]
and we are given f'(a)=5

Answer 6

f'(a)=5
means f has slope at x=a as 5

Answer 7

very good

Answer 8

Thank you!

Answer 9

It's not the slope of the secant line, btw.

Answer 10

The second one is " The slope of the tangent line to the function at x = a is 5.
"

Answer 11

That is how I read the choices

Answer 12

f(a)=5 is the first choice right?

Answer 13

What @freckles said is correct. That is the correct answer also.

Answer 14

Oh is that really the second one??? I thought f(a) = 5 was part of the question xD Lmao, sorry then

Answer 15

Oh that's what I was confused about earlier lol.

Answer 16

yeah we don't have two points so it can't be secant line

Answer 17

I'm just saying this for clarity purposes

Answer 18

\[\frac{f(x)-f(a)}{x-a} \text{ would be slope of the secant line through points } \\ (x,f(x)) \text{and } (a,f(a)) \\ \lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a} \text{ would be slope of tangent line at } x=a \]

Answer 19

Perfect!