Limits to infinity (n)

Question

Hi I have a question regarding finding the values of limit for the following equation.

The question states to find the following limits:

$$ \lim_{x\to\infty}\left(\frac {x^2+2x-1}{2x^3-3-2}\right)^\frac{1}{x} $$

Thank You!!!

Answer 1

A common trick with infinite limits is to divide both the numerator and denominator by the highest power of $x$ in the numerator. I'm assuming you meant $3x$ or $3x^2$ in the denominator; apply the same trick for whichever it is.

\begin{align} \frac{x^2 + 2x - 1}{2x^3 - 3x - 2} &= \frac{1 + 2/x - 1/x^2}{2x - 3/x - 2/x^2} \end{align}

Another trick where the variable appears in the exponent is to try taking the limit of the natural logarithm first. Let $y$ be the expression including the exponent. Then, with the hint I gave above, you can show that

\begin{equation*} \lim_{x \to \infty } 1/x \ln y = 0. \end{equation*}

See if you can get somewhere with this.