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 $\displaystyle \frac{12}{x}\ \rightarrow\ 0$
$\displaystyle \frac{\frac{1}{x}\left(x^2\right)}{\frac{1}{x}(x+1 0)}=\frac{x}{1+\frac{10}{x}}$
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 $\displaystyle \frac{\left(\frac{1}{x^2}\right)}{\left(\frac{1}{x ^2}\right)}$