  ### Home > CALC > Chapter 2 > Lesson 2.3.1 > Problem2-110

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.)

1. $\lim\limits_ { x \rightarrow \infty } ( \frac { x ^ { 2 } - 7 x + 6 } { x ^ { 3 } + 9 x - 2 } )$

Compare the highest power in the numerator and the denominator small exponent $(x^²)$ on top large exponent $(x^³)$ on the bottom.

$\lim\limits_{x\rightarrow \infty }\frac{x^{2}}{x^{3}}=\lim\limits_{x\rightarrow \infty }\frac{1}{x}$

$0$

1. $\lim\limits_ { x \rightarrow \infty } ( \frac { x ^ { 2 } - 7 x + 6 } { x ^ { 2 } + 9 x - 2 } )$

As $x→∞$, only the highest power of the numerator and denominator is powerful enough to affect the end behavior of a function.

$\lim\limits_{x\rightarrow \infty }\frac{x^{2}}{x^{2}}=$

$1$

1. $\lim\limits_ { x \rightarrow \infty } ( \frac { x ^ { 3 } - 7 x + 6 } { x ^ { 2 } + 9 x - 2 } )$

$\lim\limits_{x\rightarrow \infty }\frac{x^{3}}{x^{2}}=\lim\limits_{x\rightarrow \infty }x=$

Limit Does Not Exist but $y → ∞$