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Question
Evaluate: `lim_(x -> 1) (x^7 - 2x^5 + 1)/(x^3 - 3x^2 + 2)`
Solution
Given that `lim_(x -> 1) (x^7 - 2x^5 + 1)/(x^3 - 3x^2 + 2)` .....`[0/0 "form"]`
= `lim_(x -> 1) (x^7 - x^5 - x^5 + 1)/(x^3 - x^2 - 2x^2 + 2)`
= `lim_(x -> 1) (x^5(x^2 - 1) - 1(x^5 - 1))/(x^2(x - 1) - 2(x^2 - 1))`
Dividing the numerator and denominator by (x – 1) we get
= `lim_(x -> 1) (x^5 ((x^2 - 1)/(x - 1)) - 1((x^5 - 1)/(x - 1)))/(x^2((x - 1)/(x - 1)) - 2((x^2 - 1)/(x - 1))`
= `(lim_(x -> 1) x^5 (x + 1) - lim_(x -> 1) ((x^5 - (1)^5)/(x - 1)))/(lim_(x -> 1) x^2 - 2 lim_(x -> 1) (x + 1))`
= `(1(2) - 5 * (1)^(5 - 1))/(1 - 2(2))`
= `(2 - 5)/(1 - 4)`
= `(-3)/(--3)`
= 1
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