Thread: Limits to Infinity and Slant Asymptotes

1. Limits to Infinity and Slant Asymptotes

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

2. 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: . $y \;=\;\frac{1}{3}x - \frac{1}{3}$

3. Thank you!

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

4. help plz?

can anyone still help me with

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

5. Divide top & bottom by $x^2.$

6. Thank You

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

7. Yes.

8. YAY

THANK YOUUUUU! lol

YAY im good to go