WolframAlpha wrong evaluate: $\lim\limits_{x \rightarrow a}\frac{x^m - a ^m}{ x^n-a^n}$

Question

Let $a \in \mathbb{R}$ and $m,n\in \mathbb{N}$. If i let evaluate WolframAlpha this: $\lim\limits_{x \rightarrow a}\frac{x^m - a ^m}{ x^n-a^n}$ i get $\frac{ma^{m-n}}{n}$. In my opinion this is not true for $a = 0 $ and $n >m$.

Answer 1

If you include $\infty$ as a possible limit the Wolfram answer is Ok. The factor $m/n$ has no influence on the answer in the case $a=0$ as it is either $0,1$ or $\infty$.

(I presume you are looking at the limit: $\lim_{x\rightarrow a} \frac{x^m-a^m}{x^n-a^n}$)

Answer 2

wolfram alpha is right:$$\lim\limits_{x \rightarrow a}\frac{x^m - a ^m}{ x^n-a^n}=\lim\limits_{x \rightarrow a}\frac{mx^{m-1}}{nx^{n-1}}=\lim\limits_{x \rightarrow a}\frac{mx^{m-1-(n-1)}}{n}=\lim\limits_{x \rightarrow a}\frac{mx^{m-n}}{n}=\frac{ma^{m-n}}{n}$$ in the case where $n>m$ we get $\LARGE\frac{ma^{c}}{n}=\frac{m}{na^{-c}}$ where $c$ is a negative number. if you take this limit you get $\infty$ if $2\mid n-m$ and $\pm\infty$ elsewhere