CPM Homework Help : CALC Problem 2-110

Lena loves limits! She wants to find a shortcut for limits so she can predict a function's value without graphing. Use the following limits to look for a shortcut and explain to Lena how she can find $\lim\limits_ { x \rightarrow \infty } \frac { p ( x ) } { r ( x ) }$ without graphing. (Be careful! The expressions below are all slightly different.)

$\lim\limits_ { x \rightarrow \infty } ( \frac { x ^ { 2 } - 7 x + 6 } { x ^ { 3 } + 9 x - 2 } )$
Hint (a):
Compare the highest power in the numerator and the denominator small exponent $(x^²)$ on top large exponent $(x^³)$ on the bottom.
More Help (a):
$\lim\limits_{x\rightarrow \infty }\frac{x^{2}}{x^{3}}=\lim\limits_{x\rightarrow \infty }\frac{1}{x}$
Answer (a):
$0$

$\lim\limits_ { x \rightarrow \infty } ( \frac { x ^ { 2 } - 7 x + 6 } { x ^ { 2 } + 9 x - 2 } )$
Hint (b):
As $x→∞$ , only the highest power of the numerator and denominator is powerful enough to affect the end behavior of a function.
More Help (b):
$\lim\limits_{x\rightarrow \infty }\frac{x^{2}}{x^{2}}=$
Answer (b):
$1$

$\lim\limits_ { x \rightarrow \infty } ( \frac { x ^ { 3 } - 7 x + 6 } { x ^ { 2 } + 9 x - 2 } )$
Hint (c):
$\lim\limits_{x\rightarrow \infty }\frac{x^{3}}{x^{2}}=\lim\limits_{x\rightarrow \infty }x=$
Answer (c):
Limit Does Not Exist but $y → ∞$