Evaluate: `lim_(x->1) (x^15-1)/(x^10-1)`

1.	Evaluate: `lim_(x->1) (x^15-1)/(x^10-1)`
Answer» numerator,`x^15 - 1` `(x^5)^3 - 1` `= (x^5-1)[(x^5)^2 + x^5 + 1]` `= (x^5 - 1)[x^10 + x^5 +1]` denominator,. `x^10 - 1 = (x^5)^2 - 1` `= (x^5 - 1)(x^5+1)` `lim_(x->1) ((x^5 -1){x^10 + x^5 +1})/((x^5 - 1)(x^5+1))` `lim_(x->1) (x^10 + x^5 +1)/x^5` `= (1^10 + 1^5+ 1)/(1^5 +1)= 3/2` `lim_(x->1) (x^15 - 1)/(x^10 - 1)` `= lim_(x->1) (d/dx(x^15-1))/(d/dx(x^10-1))` `lim_(x->1) (15x^14)/(10 x^9)` `= 15/10 = 3/2` Answer

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))`
Evaluate, `lim_(x to 1) (x^(4)-1)/(x-1)=lim_(x to k) (x^(3)-k^(3))/(x^(2)-k^(2))` , then find the value of k.
Evaluate: `lim_(x->1) (x^15-1)/(x^10-1)`
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)`