On one of my homework problems it's asking for me to sketch a graph of a function which fulfills five requirements. All of the requirements make sense to me other than the following:
$$\lim_{x\rightarrow \ \infty} f(x) = {-\infty}$$
How could there be a limit approaching infinity at negative infinity? Doesn't one of the number need to be finite for me to know where to place the infinity limit? This makes no sense to me and is really throwing me off.
$$f'(x) \ge 2 \ \text{when} -2 \lt x \lt 0 \ \text{and} \ \text{when} \ 2 \lt x \lt 4 $$
I don't even know how to approach this one. How do you apply information about the derivative/tangent line to the graph of the original equation?
Other requirements (purely for reference, I understand these):
$$\lim_{x\rightarrow \ 0+} f(x) = -2$$ f(x) is an even function
f(x) is differentiable everywhere but two x-values, and is continuous for all x
Thank you.
Saying that $$\lim\limits_{x\to\infty}f(x)=-\infty$$ just means that for each real number $M$ there is an $x_M$ such that $f(x)<M$ whenever $x\ge x_M$. Taking $M=-100$, for instance, there is an $x_{-100}$ such that $f(x)<-100$ whenever $x\ge x_{-100}$. A very simple example of such a function is the function $f(x)=-x$: for this function you can take $x_{-100}$ to be $101$ for instance, since $f(x)=-x<-100$ whenever $x\ge 101$.
For the other requirement, remember that $f\,'(a)$ is the slope of the tangent line to the curve $y=f(x)$ at $x=a$. The requirement that $f\,'(x)\ge 2$ when $2<x<4$ just means that the graph has a slope of at least $2$ over the entire interval $(2,4)$. Many functions have this property: $f(x)=ax$ with $a\ge 2$ (since $f\,'(x)=a$), $f(x)=x^2$ (since $f\,'(x)=2x$, which is certainly greater than $2$ when $2<x<4$, and many, many more.
Remember, you don’t have to write down a single expression for $f$: you can define it piecewise, with different definitions on different intervals.