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Let $0 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!

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    Be careful. Can you use L'Hopital? Do you really get "0/0"? Also, even if L'Hopital applies, you must differentiate with respect to "t" not "x" ("t" is the limiting variable).2011-10-06

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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 $+\infty$.

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    I want to learn the difference between say the limit is $+\infty$ and saying that the limit does not exist. Thanks.2011-10-06