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Question
Evaluate: `lim_(x -> 1) (x^4 - sqrt(x))/(sqrt(x) - 1)`
Solution
Given that `lim_(x -> 1) (x^4 - sqrt(x))/(sqrt(x) - 1)`
= `lim_(x -> 1) (sqrt(x)[(x)^(7/2) - 1])/(sqrt(x) - 1)`
= `lim_(x -> 1) (sqrt(x) ([x^(7/2) - (1)^(7/2)])/(x - 1))/(((x)^(1/2) - (1)^(1/2))/(x - 1))` .....[Dividing the numerator and denominator of x – 1]
= `lim_(x -> 1) (((x)^(7/2) - (1)^(7/2))/(x - 1))/(((x)^(1/2) - (1)^(1/2))/(x - 1)) xx lim_(x -> 1) sqrt(x)` .....`[because lim_(x -> a) f(x) g(x) - lim_(x -> a) f(x) * lim_(x -> a) g(x)]`
= `(7/2 (1)^*7/2 - 1)/(1/2(1)^(1/2 - 1)) xx sqrt(1)`
= `(7/2)/(1/2)`
= 7
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