  ### Home > APCALC > Chapter 3 > Lesson 3.3.1 > Problem3-95

3-95.

Evaluate each limit. If the limit does not exist due to a vertical asymptote, then add an approach statement stating if y is approaching negative or positive infinity.

1. $\lim\limits _ { x \rightarrow 0 } \large\frac { x ^ { 2 } + 3 x - 10 } { x - 2 }$

This limit can be evaluated without any fancy Algebra steps.

1. $\lim\limits _ { x \rightarrow - 5 } \large\frac { x ^ { 2 } + 3 x - 10 } { x - 2 }$

Refer to the hint in part (a). Limits like these suggest that the actual value $f(−5)$ and the limit $x\rightarrow −5$ agree.

1. $\lim\limits _ { x \rightarrow 2 } \large\frac { x ^ { 2 } + 3 x - 10 } { x - 2 }$

Factor first. Since the denominator $\rightarrow 0$, we are investigating whether there is a hole or an asymptote.

1. $\lim\limits _ { x \rightarrow \infty } \large\frac { x ^ { 2 } + 3 x - 10 } { x - 2 }$

This limit $\rightarrow \infty$.Compare the highest-power term in the numerator and denominator. Does the graph have a horizontal asymptote or does it approach $\infty$ or $-\infty$?

1. Use the limits above to describe the shape of the graph of $y = \large\frac { x ^ { 2 } + 3 x - 10 } { x - 2 }$. State all horizontal asymptotes, vertical asymptotes, and holes.

Refer to the Hints in parts (c) and (d).