Nrthugu

I need help to find the limits of these two functions :
$$lim_{xto a}frac{x^n-a^n}{x^p-a^p}$$ where $n,p$ are integers.

And :
$$lim_{xto a}{frac{xsin(a)-asin(x)}{x-alog_a(x)}}$$ where $a>0$ and $anotin{{1,e}}$

I can't use L'Hospital rule, any help would be very appreciated !

$$frac{x^n-a^n}{x^p-a^p}=frac{x^n-a^n}{x-a}cdotfrac{x-a}{x^p-a^p}xrightarrow[xto a]{}(x^n)'|_{x=a}cdotfrac1{(x^p)'|_{x=a}}=frac{na^{n-1}}{pa^{p-1}}=frac npa^{n-p}$$

HINT

By $f(x)=x^n$ and $g(x)=x^p$ we have

$$lim_{xto a}frac{x^n-a^n}{x^p-a^p}=lim_{xto a}frac{x^n-a^n}{x-a}frac{x-a}{x^p-a^p}=frac{f'(a)}{g'(a)}$$

and by $f(x)=xsin(a)-asin(x)$ and $g(x)=a-alog_a(x)$ we have

$$lim_{xto a}{frac{xsin(a)-asin(x)}{a-alog_a(x)}}=lim_{xto a}{frac{xsin(a)-asin(x)}{x-a}}{frac{x-a}{a-alog_a(x)}}=frac{f'(a)}{g'(a)}$$

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.

The first one simply obtain by L"Hospital rule:
$$lim_{xto a}dfrac{x^n-a^n}{x^p-a^p}=lim_{xto a}dfrac{nx^{n-1}}{px^{p-1}}=dfrac{n}{p}a^{n-p}$$

Edit:
By identity
$$lim_{xto a}dfrac{x^n-a^n}{x^p-a^p}=lim_{xto a}dfrac{(x-a)(x^{n-1}+ax^{n-2}+a^2x^{n-3}+cdots+a^{n-2}x+a^{n-1}}{(x-a)(x^{p-1}+ax^{p-2}+a^2x^{p-3}+cdots+a^{p-2}x+a^{p-1}}=dfrac{n}{p}a^{n-p}$$

For the second limit, we can directly say that it is not indeterminate form so directly put x = a.
We get required limit = $frac{a sin(a) - a sin(a)}{a-1} = 0$

Hope it helps:)