Limits to Infinity and Slant Asymptotes

**Khaali91** · October 2nd 2008, 02:45 PM

1. find lim(as x approaches positive infinity) of
2x+xsin(3x)
5x^2-2x+1

2. Find the horizontal,vertical, and slant asymptotes of:
y= x^3-X^2+10
3x^2-4X

Any help would be greatly appreciated, Thank YOU

**Soroban** · October 2nd 2008, 03:08 PM

Hello, Khaali91!

1. Find the horizontal, vertical, and slant asymptotes of:

$y \;=\;\frac{x^3-x^2+10}{3x^2-4x}$

For a horizontal asymptote, consider $\lim_{x\to\infty} y$

Divide top and bottom by $x^2\!:\quad \frac{\dfrac{x^3}{x^2} - \dfrac{x^2}{x^2} + \dfrac{10}{x^2}}{\dfrac{3x^2}{x^2} - \dfrac{4x}{x^2}} \;=\;\frac{x - 1 + \dfrac{10}{x^2}}{3-\dfrac{4}{x}}$

Then: . $\lim_{x\to\infty}\frac{x - 1 + \frac{10}{x^2}}{3-\frac{4}{x}} \;=\; \frac{\infty - 1 + 0}{3 - 0} \;=\;\infty$

. . There is no horizontal asymptote.

For vertical asymptotes, determine when the denominator equals zero.

We have: . $x(3x+4) \:=\:0\quad\Rightarrow\quad x \;=\;0,\:\frac{4}{3}$

The vertical asymptotes are: . $x\:=\:0\text{ (y-axis)},\;x \:=\: \frac{4}{3}$

For slant asymptotes, consider $\lim_{x\to\infty} y$

We have: . $\lim_{x\to\infty}\frac{x^3 - x^2 + 10}{3x^2-4x} \;=\;\lim_{x\to\infty}\frac{x - 1 + \frac{10}{x^2}}{3 - \frac{4}{x}} \;=\;\frac{x - 1 + 0}{3 - 0} \;=\;\frac{x-1}{3}$

Therefore, the slant asymptote is: . $y \;=\;\frac{1}{3}x - \frac{1}{3}$

**Khaali91** · October 2nd 2008, 03:12 PM

For the 1st one where you said the assume the limit is to infinity for the horizontal asymptote, can that be applied to all horizontal asymptotes?

THANK YOU

**Khaali91** · October 2nd 2008, 03:52 PM

can anyone still help me with

1. find the limit (as x approaches positive infinity) of
2x+xsin(3x)
5x^2-2x+1

**Krizalid** · October 2nd 2008, 04:02 PM

Divide top & bottom by $x^2.$

**Khaali91** · October 2nd 2008, 04:30 PM

but would the limit of sin3x/x still equal out to 0? thats where i got confused =/

**Krizalid** · October 2nd 2008, 04:40 PM

Yes.

**Khaali91** · October 2nd 2008, 04:44 PM

THANK YOUUUUU! lol

YAY im good to go

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