Limits of functions as x approaches a I need help to find the limits of these two functions :
$$\lim_{x\to a}\frac{x^n-a^n}{x^p-a^p}$$ where $n,p$ are integers.
And : 
$$\lim_{x\to a}{\frac{x\sin(a)-a\sin(x)}{x-a\log_a(x)}}$$ where $a>0$ and $a\notin{{1,e}}$
I can't use L'Hospital rule, any help would be very appreciated !
 A: $$\frac{x^n-a^n}{x^p-a^p}=\frac{x^n-a^n}{x-a}\cdot\frac{x-a}{x^p-a^p}\xrightarrow[x\to a]{}(x^n)'|_{x=a}\cdot\frac1{(x^p)'|_{x=a}}=\frac{na^{n-1}}{pa^{p-1}}=\frac npa^{n-p}$$
A: HINT
By $f(x)=x^n$ and $g(x)=x^p$ we have
$$\lim_{x\to a}\frac{x^n-a^n}{x^p-a^p}=\lim_{x\to a}\frac{x^n-a^n}{x-a}\frac{x-a}{x^p-a^p}=\frac{f'(a)}{g'(a)}$$
and by $f(x)=x\sin(a)-a\sin(x)$ and $g(x)=a-a\log_a(x)$ we have
$$\lim_{x\to a}{\frac{x\sin(a)-a\sin(x)}{a-a\log_a(x)}}=\lim_{x\to a}{\frac{x\sin(a)-a\sin(x)}{x-a}}{\frac{x-a}{a-a\log_a(x)}}=\frac{f'(a)}{g'(a)}$$
A: The first one simply obtain by L"Hospital rule:
$$\lim_{x\to a}\dfrac{x^n-a^n}{x^p-a^p}=\lim_{x\to a}\dfrac{nx^{n-1}}{px^{p-1}}=\dfrac{n}{p}a^{n-p}$$
Edit:
By identity
$$\lim_{x\to a}\dfrac{x^n-a^n}{x^p-a^p}=\lim_{x\to a}\dfrac{(x-a)(x^{n-1}+ax^{n-2}+a^2x^{n-3}+\cdots+a^{n-2}x+a^{n-1}}{(x-a)(x^{p-1}+ax^{p-2}+a^2x^{p-3}+\cdots+a^{p-2}x+a^{p-1}}=\dfrac{n}{p}a^{n-p}$$
A: For the first one you can factor top and bottom and cancel $(x-a)$
For the second one the answer is straight forward because the top goes to zero and the bottom does not. 
