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Question
Evaluate the following limit:
`lim_(x -> - 2)[(x^7 + x^5 + 160)/(x^3 + 8)]`
Solution
`lim_(x -> - 2)[(x^7 + x^5 + 160)/(x^3 + 8)]`
= `lim_(x -> - 2) ((x^7 + 128) + (x^5 + 32))/(x^3 + 8)`
= `lim_(x-> - 2) (((x^7 + 128) + (x^5 + 32))/(x + 2))/((x^3 + 8)/(x + 2)) ...[("As" x -> - 2"," x ≠ -2),(therefore x + 2 ≠ 0),("Divide Numerator and"),("Denominator by" x + 2)]`
= `(lim_(x -> - 2) ((x^7 + 2^7)/(x + 2) + (x^5 + 2^5)/(x + 2)))/(lim_(x -> - 2)((x^3 + 2^3)/(x + 2))`
= `(lim_(x -> - 2) (x^7 - ( - 2)^7)/(x - ( - 2)) + lim_(x -> - 2) (x^5 - ( - 2)^5)/(x - ( - 2)))/(lim_(x -> - 2) (x^3 - ( - 2)^3)/(x - ( - 2))`
= `(7 (-2)^6 + 5(-2)^4)/(3 (-2)^2) ...[lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `(7(64) + 5(16))/(3(4)`
= `(448 + 80)/12`
= `528/12`
= 44
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