If the limit :?

Question

lim f(3+h)-f(3)/ 3
h->0
exists, finds its value
A) lim=7
B) lim=23
C) lim= -9
D) lim= -1
E) limit doesnt exist

Let f be the function defined by f(x)=7x+(x-5)+ lx-5l)^2

Answer 1

The question is
lim f(3+h)-f(3)/ h
h->0

not what you have typed.
Then,
lim f(3+h)-f(3)/ h = f'(3)
h->0

For knowing f'(3) you must supply the function f(x)

Now that you have given f(x), the problem can be solved.

f(x) = 7x + (x - 5) + lx - 5l^2

f(3) = 7*3 + (3 - 5) + l3 - 5l^2
= 21 + (-2) + |-2|^2 = 21 - 2 + 4 = 23

f(h + 3) = 7(h + 3) + (h + 3 - 5) + lh + 3 - 5l^2
= 7(h + 3) + (h - 2) + (h - 2)^2
= 7h + 21 + h - 2 + h^2 - 4h + 4
= h^2 + 4h + 23

Then, the given limit
lim f(3+h)-f(3)/ h
h->0

= lim [(h^2 + 4h + 23) - (23)]/ h
h->0

= lim (h^2 + 4h)/ h
h->0

= lim (h + 4)
h->0

= 4 .... not any of the answer you provided

However, the limit for what you typed is zero.

Answer 2

As others have said, you need to specify something about f.

If f is continuous at 3, then the limit as typed above is 0. If f is discontinuous at 3 we can achieve any of the answers listed above. (For answers A through D, take f(3) = 0, f(x) = desired limit * 3 when x ≠ 3. For answer E, take f(3) = 0, f(x) = 1 for x > 3, f(x) = -1 for x < 3.)

As others have said, it is likely that this should be the limit of (f(3+h)-f(3) / h, i.e. f'(3), but we can't evaluate that without knowing f.

Answer 3

lim f(3+h)-f(3)/ 3
h->0
As h ---> 0, we get f(3) -f(3)/3 = 2f(3)/3.
The above is not one of the answers, so perhaps you meant
lim [f(3+h)-f(3)]/ 3
h->0
In this case the limit is zero but it is also not among the answers.
So perhaps you meant the difference quotient which is:
lim [f(3+h)-f(3)]/h
h->0
In this case the limit is 0/0 which is indeterminate meaning the
limit might or might not exist. The best answer among those given would be E.

Answer 4

You need to specify what the function f is before anyone can help.