
Functions and Limits
Hi can you check if these answers are correct?
1. Sketch the graph of a function that satisfies these conditions:
limf(x) = 2
x>0
lim f(x) = 0
x>0+
limf(x)=3
x>4
limf(x)=0
x>4+
f(0) = 2
f(4) = 1
Does this give me the these points?
(0,2) approaching from the left
(0,0) approaching from the right
(4,3) approaching from the left
(4,0) approaching from the right
(0,2)
(4,1)
Also, since I have the point (0,2) twice, is there anything special I need to do with it?
2. Determine the infinite limit
lim (x^{2} + 1) / (x^{2}  4x + 4)
x>2
= (x(x2)) / (x2)(x2)
= x / (x2)
=2 / 0 = positive infinity
3. Graph this function
y=(x^{2}+1) / (3x2x^{2})
How do i graph this without using a calculator?

Re: Functions and Limits
For (1), you need a "solid dot" at (0,2) and an "open circle" at (0,0). Similarly, you need a "solid dot" at (4,1) and "open circles" at (4,0) and (4,3).
In problem (2) you factored $\displaystyle x^2+1$ to get $\displaystyle x(x2)$, which is incorrect. You only need to factor the denominator to get $\displaystyle \frac{x^2+1}{(x2)^2}$. The answer is still $\displaystyle \infty$.
For (3), $\displaystyle 3x2x^2$ has two zeros, which will be vertical asymptotes of the function. As x goes to $\displaystyle +\infty$ or $\displaystyle \infty$, y goes to $\displaystyle \frac{1}{2}$. Beyond that, I think you might have to calculate a few points. A graphing calculator or graphing tool on the computer is good for checking your work.
 Hollywood