1. ## Limits

Consider f (x) = (-7x^3 + 2)/(10x^3 + 3x + 2)

a) Construct a table of values to determing lim f(x) x to infinity

b) Use algebra to determin lim f(x)
x to infinity

c) Sketch the graph and indicate on the graph the behavior described by the limit.

2. part a. plug and chug.

For part b, get rid of everything but the highest powers in the num. and den.

You are left with: $\displaystyle \frac{-7x^{3}}{10x^{3}}$

Now, you can see what the limit is, can't you?.

part c. You have a graphing calculator?.

3. This is my 18th Post!!!

4. Originally Posted by galactus
part a. plug and chug.

For part b, get rid of everything but the highest powers in the num. and den.

You are left with: $\displaystyle \frac{-7x^{3}}{10x^{3}}$

Now, you can see what the limit is, can't you?.

part c. You have a graphing calculator?.
Umm am not really sure I understand what you meant y plug and chug.
Can I plug in any value that I like? Arent I supposed to follow a sequence or something??

For part B how can I just get rid of everything but the highest powers?
L'Hospital Rule??

5. Just plug in higher and higher values and watch it approach the limit.

$\displaystyle \frac{-7(1)^{3}+2}{10(1)^{3}+3(1)+2}=\frac{-1}{3}$

$\displaystyle \frac{-7(100)^3+2}{10(100)^{3}+3(100)+2}=\frac{-3499999}{5000151}=-.699978660644$

$\displaystyle \frac{-7(1000)^{3}+2}{10(1000)^{3}+3(1000)+2}=\frac{-3499999999}{5000001501}=-.69999978966$

See, it's approaching $\displaystyle -.7\;\ or\;\ \frac{-7}{10}$

Now, make it official with part b. Rational functions, especially with polynomials, are one of the easiest things in which to take a limit.

For part b, no, just erase them. In a rational function, if the limit approaches infinity, it is unaffected if you discard everything but the highest powers. Think about it. See?.

Suppose I had $\displaystyle \lim_{x\to\infty}\frac{3x+5}{6x-8}$

If x gets huge, towards infinity, what are the 5 and 6 going to amount to?. Nothing. So, get rid of them and take the limit of 3x/6x. Which is 3/6=1/2.

Yours works the same.

$\displaystyle \frac{-7x^{3}}{10x^{3}}$

$\displaystyle \frac{-7x^3 + 2}{10x^3 + 3x + 2}$
Divide numerator and denominator by $\displaystyle x^3$,
$\displaystyle \frac{-7+\frac{2}{x^3}}{10+\frac{3}{x^2}+\frac{2}{x^3}} \to -\frac{7}{10}$ as, $\displaystyle x\to\infty$
$\displaystyle \frac{k}{x^n}\to 0$