Evaluate `lim_(xtoa) (sinx-sina)/(sqrt(x)-sqrt(a))`

1.	Evaluate `lim_(xtoa) (sinx-sina)/(sqrt(x)-sqrt(a))`
Answer» Given, `underset(xtoa)"lim"(sinx-sina)/(sqrt(x)-sqrt(a)) =underset(xto0)"lim"((2cos(x+a))/2sin(x-a)/2)/(sqrt(x)-sqrt(a))`<br> `[therefore sinC-sinD=2cos(C+D)/2.sin(C-D)/2]`<br> `=underset(xto0)"lim"((2cos(x+a))/2sin(x-a)/2(sqrt(x)+sqrt(a)))/((sqrt(x)-sqrt(a))(sqrt(x)+sqrt(a)))`<br> `=underset(xto0)"lim"((2cos(x+a))/2sin(x-a)/2 (sqrt(x)+sqrt(a)))/(x-a)`<br> `=2underset(xto0)"lim"cos(x+a)/(2)(sqrt(x)+sqrt(a)).1/2underset(xto0)"lim"(sin((x-2)/2))/(2((x-a)/2))`<br> `=2underset(xto0)"lim"cos((x+a)/2)(sqrt(x)+sqrt(a)). 1/2underset(xto0)"lim"(sin((x-a)/2))/((x-a)/2)`<br> `=2.cosa/2.sqrt(a).1/2` `[therefore underset(xto0)"lim"(sinx)/x=1]`<br> `=sqrt(a)cosa/2`

Discussion

Differentiate function with respect to x,f(x)=`(x^(4)+x^(3)+x^(2)+1)/(x)`
Evaluate `lim_(xtoa) (sinx-sina)/(sqrt(x)-sqrt(a))`
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Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(4x+5sinx)/(3x+7cos x)`