Let $0<x<1$ be a real number and let $a$ be a positive integer.
I want to compute $$\lim_{t\to 0} \frac{ (x+\frac{t}{x^2})^a}{t^a}.$$
The only thing I can come up with is to use L'Hopital's rule.
Differentiating the denominator and numerator $a$ times I end up with
$$\lim_{t\to 0} \frac{ (x+\frac{t}{x^2})^a}{t^a} = \frac{1}{x^{2a}}.$$
Is this correct?
Thnx for your help!
Best Answer
Since $x$ is fixed and $t\to0$, $x+t/x^2\to x$ hence $(x+t/x^2)^a$ converges to $x^a$. The ratio $\dfrac1{t^a}$ converges to $\pm\infty$, depending on whether $t\to0^+$ or $t\to0^-$, and on the parity of the positive integer $a$, hence the whole thing converges to $\pm\infty$ as well.
If $a$ is even, the limit is $+\infty$. If $a$ is odd, the limit does not exist, but the limit from the left (that is, when $t\to0^-$) is $-\infty$ and the limit from the right (that is, when $t\to0^+$) is $+\infty$.