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.
Umm am not really sure I understand what you meant y plug and chug.Originally Posted by galactus
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??
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.
$\displaystyle \frac{-7x^{3}}{10x^{3}}$
You have,Originally Posted by askmemath
$\displaystyle \frac{-7x^3 + 2}{10x^3 + 3x + 2}$
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, $\displaystyle x\to\infty$
Becuase,
$\displaystyle \frac{k}{x^n}\to 0$