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

Discussion

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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)`