how do i find a formula for a function " f " that satisfies certain (infinite) limit conditions?

Question

anonymous · Answer

$$\lim_{x \rightarrow 0}=(f(x+\Delta x)−f(x))/\Delta x$$

anonymous · Answer

this is an example of what i am asking about

anonymous · Answer

#49

zarkon · Answer

look for a rational function that works.    It is not a complicated function end the end

kmeis002 · Answer

Recall That 
$$\lim_{x \rightarrow \pm \infty} \frac{ax^{n}+...}{bx^{m}+...}  $$
Where they are rational functions, recall that so long as m>n, then the limit at infinities will equal zero.

The other conditions depend on the denominator and numerator, respectively. So try to come up with a vertical asymptote at x = 0 (what does that mean for the denominator?)

And how would you get f(2) = 0? (think about the numerator)

anonymous · Answer

thank you soo much :)))

anonymous · Answer

i found the answer but i have one question :/

anonymous · Answer

the purple box is the real answer, but i dont know where the negative (-) comes from

kmeis002 · Answer

Do you mean the 2-x?

anonymous · Answer

yeah

kmeis002 · Answer

For the condition $$ f(2) = 0$$

Implies the numerator must be zero so it must have a factor that = 0.

When x = 2 then (2-x) = 0

anonymous · Answer

but doesnt (x-2) also equal two

anonymous · Answer

they both equal to 2

kmeis002 · Answer

Ah, it arises from the fact that $$ \lim_{x \rightarrow 0} f(x) = -\infty$$

(x-2) when x approaches zero results in a negative
(2-x) when x approaches zero results in a positive

And since (x-3) results in a negative, your factor up top must result in a positive or else 
 $$ \lim_{x \rightarrow 0} f(x) = \infty$$

anonymous · Answer

OH OKAY! thank you soo much