Finding f(x) limits using given limit conditions

[aleferesco]

1) If the limit of the function g(x) = [f(x) -8]/[x-1] is equals to 10 as x approches 1, then find the limit of f(x) as x approaches 1


2) If the limit of the function h(x) = f(x)/x² is equals to 5 as x approaches 0, then find the following limits

a) limit of f(x) as x approaches zero

b) limit of f(x)/x as x approaches zero


---> Attempt answer for question number 1

..so what I understand from the two questions is that they don't really require much calculations, These only require the understanding of what's happening to the original limits. I know that in the first question the fact that they are tellings us that there is a limit for g(x) helps us determine that there has to be also a limit for f(x) since it is part of the g(x). So basically I know that f(x) has to be at least 8 so that when I substitute the x=1 values into g(x), it will give me an indeterminate number (0/0)

so for some reason the answer to this question is: the limit of of f(x) is equals to 8 as x approaches 1 (does the explanation that I made supports this answer, if not please help)


---> Attempt answer for question number 2

don't really know how to start to solve, please help



ps. If you are confused by my attempted explanation to question number 1, please let me know or explain it to me in the way that you understand it

ps2. Sorry that I gave an verbal representation for the questions instead of writting the equations out (I'm new in this forum)


thanks

 

[gabbagabbahey]

I think to properly solve the first question you need to exploit some of the properties of limits.
For example, if the limits as x->c of u(x) and v(x) exist and are finite, the following hold true:

$$\lim_{x \to c} u(x)v(x)= \left( \lim_{x \to c} u(x) \right) \left( \lim_{x \to c} v(x) \right)$$

and

$$\lim_{x \to c} (u(x)-v(x))= \left( \lim_{x \to c} u(x) \right) - \left( \lim_{x \to c} v(x) \right)$$

Do you see how these properties help you here?

 

[Dick]

You have the right idea. Sure lim x->0 f(x)=8, in the first case. If the limit were not 8 then the limit of the quotient would not exist. Apply the same reasoning to the second. x^2->0. What must f(x) approach? 1/x goes to infinity. What must f(x)/x approach?