Evaluate the limit of the fraction (f(x)-f(1))/(x-1), if f(x)=1+2x^5/x^2?

x->1

Expert Answers

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We are given that f(x)=1+2x^5/x^2 = 1 + 2x^3.

We have to find:  lim x -->1 [(f(x) - f(1))/(x-1)]

=> lim x -->1 [(1+ 2x^3 - 1 - 2)/(x-1)]

=> lim x -->1 [(2x^3 - 2)/(x-1)]\

=> lim x -->1 [2*(x - 1)(x^2 + x + 1)/(x-1)]

=> lim x -->1 [2*(x^2 + x + 1)]

substitute x with 1

=> 2*(1 +1 +1)

=> 6

Therefore the required limit  is 6.

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