`lim_(x->1) (x^a - ax + a - 1)/(x - 1)^2` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule...

`lim_(x->1) (x^a - ax + a - 1)/(x - 1)^2` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.

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Chapter 4, 4.4 - Problem 37 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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nees101 | Student, Graduate | (Level 2) Adjunct Educator

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Given the limit `lim_{x->1}(x^a-ax+a-1)/(x-1)^2` . We have to find the limit value.

Applying the limits we get,

`lim_{x->1}=(x^a-ax+a-1)/(x-1)^2=(1^a-a+a-1)/(1-1)^2=0/0`      Since `1^a=1`

So using L'Hospital's rule and applying the limit we get,

`lim_{x->0}(ax^(a-1)-a)/(2(x-1))=0/0`

So again applying L'Hospital's rule we have,

`lim_{x->1}(a(a-1)x^(a-2))/2=(a(a-1))/2`

hence the limit is `(a(a-1))/2`

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