assume that f(x) is everywhere continuous and it is given to you that f(2)=-7 and
lim (f(x)+7)/(x-2)=8
x➟ 2
it follows that y = ?
I dont really understand how to solve this question as if you were to directly sub in x=2 then you get a zero on top which would leave the left side equal to 0/0=8? normally in other questions you would factor but with this you cant factor it.
thanks for any help
Given that the limit is 8, we really do need 0/0, otherwise the fraction goes to ∞. So y = -7. So, if f(2) is anything but -7, such as 3, then we have 10/0 which = ∞ -- but we are told that the limit is 8.
But they just told us that f(-2) = -7. So why ask what y is? Isn't y=f(x)?
assuming no miscommunication, this question is incredibly confusing. i think that what it might be asking is not the limit but what would make it equal to 8? so I would just plug in random numbers till I get the limit = 8
it is confusing. But since the limit is a finite value, the fraction must be 0/0. For, if 0/0 = n, then that means that 0 = n*0. But that is true for any finite n. So if they tell you that the limit is 8, the fraction must be 0/0, since the bottom is zero.
To solve this problem, we can use the fact that when the limit of a function as x approaches a certain value exists, and the function is continuous at that value, the limit is equal to the value of the function at that point.
Given that lim[(f(x)+7)/(x-2)] as x approaches 2 is 8, we can rewrite the limit as x approaches 2 as:
lim[(f(x)+7)/(x-2)] as x approaches 2 = 8
Now, if we directly substitute x = 2, we get:
[(f(2)+7)/(2-2)] = 8
Note that the function f(x) is continuous at x = 2, so the limit exists. However, as you mentioned, directly substituting x = 2 results in an indeterminate form (0/0).
To resolve this, we can rewrite the equation and use the fact that f(x) is continuous at x = 2:
lim[(f(x)+7)/(x-2)] as x approaches 2 = lim[(f(x)+7)/(x-2)] as x approaches 2
= lim[(f(x)-f(2))/(x-2)] as x approaches 2
By applying the limit, we can rewrite the equation as:
(f(2)+7)/lim[(x-2)] as x approaches 2 = 8
Since the limit of (x - 2) as x approaches 2 is 0, we have:
(f(2)+7)/0 = 8
Notice that dividing by 0 is undefined. Therefore, for this equation to hold true, the numerator (f(2) + 7) must also be equal to 0.
f(2) + 7 = 0
Now, we can solve for f(2):
f(2) = -7
So, the value of y (f(2)) is -7.