# Help with a limit question

#### rowdy3

Please determine the following limit if they exist. If the limit doe not exist put DNE.
lim 2x^3 / x^2 + 10x - 12
x->infinity.
Thanks.

#### Anonymous1

Please determine the following limit if they exist. If the limit doe not exist put DNE.
lim 2x^3 / x^2 + 10x - 12
x->infinity.
Thanks.
Do you know L'Hospitals?

$$\displaystyle \lim_{x\to\infty} \frac{2x^3}{x^2 + 10x - 12}$$

$$\displaystyle = \lim_{x\to\infty} \frac{6x^2}{2x + 10}$$

$$\displaystyle = \lim_{x\to\infty} \frac{12x}{2} = \lim_{x\to\infty} 6x \to\infty$$ $$\displaystyle \fbox{DNE}$$

• rowdy3

#### TKHunny

You may wish to review your "Order of Operations" and add a few more parentheses, but other than that, with rational functions, degree of numerator greater than degree of denominator ==> Definitely DNE.

• rowdy3

#### skeeter

MHF Helper
Please determine the following limit if they exist. If the limit doe not exist put DNE.
lim 2x^3 / x^2 + 10x - 12
x->infinity.
Thanks.
very straight-forward limit ... what do you think?

• rowdy3

Please determine the following limit if they exist. If the limit doe not exist put DNE.
lim 2x^3 / x^2 + 10x - 12
x->infinity.
Thanks.
Hi rowdy3,

As the numerator contains a higher power of x, then the expression
increases without bound as x does, hence the limit DNE.

$$\displaystyle \frac{x^3}{x^2+10x-12}=\frac{\frac{1}{x}\left(x^3\right)}{\frac{1}{x}\left(x^2+10x-12\right)}$$

$$\displaystyle =\frac{x^2}{x+10-\frac{12}{x}}$$

As $$\displaystyle x\ \rightarrow\ \infty$$ the term $$\displaystyle \frac{12}{x}\ \rightarrow\ 0$$

$$\displaystyle \frac{\frac{1}{x}\left(x^2\right)}{\frac{1}{x}(x+10)}=\frac{x}{1+\frac{10}{x}}$$

As $$\displaystyle x\ \rightarrow\ \infty$$ the term $$\displaystyle \frac{10}{x}\ \rightarrow\ 0$$

$$\displaystyle \lim_{x\ \rightarrow\ \infty}\frac{x}{1}$$ DNE

If you like, do it in one step, eliminating x from the denominator by multiplying by $$\displaystyle \frac{\left(\frac{1}{x^2}\right)}{\left(\frac{1}{x^2}\right)}$$

• rowdy3