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 \(\displaystyle \frac{\left(\frac{1}{x^2}\right)}{\left(\frac{1}{x^2}\right)}\)