How do you find the limit of # (x^2 -sqrt x)/(sqrt x -1)# as x approaches 1?

1 Answer
mason m
Nov 6, 2016

#lim_(xrarr1)(x^2-sqrtx)/(sqrtx-1)=3#

Explanation:

Note that this is in the indeterminate form #0/0#, which means we can apply L'Hopital's rule. This means the limit will be equivalent if we take the derivative of the numerator over the derivative of the denominator.

#lim_(xrarr1)(x^2-sqrtx)/(sqrtx-1)#

Apply L'Hopital's:

#=lim_(xrarr1)(d/dx(x^2-sqrtx))/(d/dx(sqrtx-1))=lim_(xrarr1)(2x-1/(2sqrtx))/(1/(2sqrtx))#

Simplifying:

#=lim_(xrarr1)(2x-1/(2sqrtx))(2sqrtx)=lim_(xrarr1)(4x^(3/2)-1)#

Now evaluating the limit:

#=4(1)-1=3#

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