lim[x:81,(x/(1+sqrtx) - 81/10) / (x-81)]

To find the limit of the given expression, we can start by substituting the limit value into the expression. In this case, the limit value is x = 81.

Substituting x = 81 into the expression, we get:

lim[x→81] [(x/(1+sqrt(x)) - 81/10) / (x-81)]

= [(81/(1 + sqrt(81))) - 81/10] / (81 - 81)

= [(81/10) - 81/10] / 0

= 0/0

We have obtained an indeterminate form of 0/0, which means we need to apply limit properties or mathematical manipulation to simplify the expression further and evaluate the limit.

One way to simplify the expression is by rationalizing the numerator. To do this, we can multiply both the numerator and denominator by the conjugate of the numerator, which is (1 - sqrt(x)).

lim[x→81] [(x/(1+sqrt(x)) - 81/10) / (x-81)]

= [(x/(1+sqrt(x)) - 81/10) / (x-81)] * [(1 - sqrt(x))/(1- sqrt(x))]

= [(x - 81*(1 - sqrt(x))/(1+sqrt(x))/(1- sqrt(x)))] / [(x-81)*(1 - sqrt(x))/(1- sqrt(x))]

= [x - 81*(1 - sqrt(x))]/[(x-81)*(1 - sqrt(x))]

Now, let's simplify further:

lim[x→81] [x - 81*(1 - sqrt(x))]/[(x-81)*(1 - sqrt(x))]

Since x = 81, the expression further simplifies to:

= [81 - 81*(1 - sqrt(81))]/[(81-81)*(1 - sqrt(81))]

= [81 - 81*(1 - 9)]/[0*(1 - 9)]

= [81 - 81*(8)]/[(0)*(-8)]

= 0/0

We still have an indeterminate form of 0/0. To evaluate the limit further, we can try applying L'Hopital's Rule, which states that for an indeterminate form 0/0, we can take the derivative of the numerator and denominator with respect to x, and again evaluate the limit.

Differentiating the numerator and denominator:

lim[x→81] [(d/dx[x - 81*(1 - sqrt(x))]) / (d/dx[(x-81)*(1 - sqrt(x))])]

= lim[x→81] [(1 - 81/(2*sqrt(x))) / (1 - sqrt(x) - (x - 81)/(2*sqrt(x)))]

Now, substituting x = 81 into the derivative expression:

= [(1 - 81/(2*sqrt(81))) / (1 - sqrt(81) - (81 - 81)/(2*sqrt(81)))]

= [(1 - 81/18) / (1 - 9 - 0/18)]

= [(1 - 9/2) / (1 - 9)]

= [-7/2] / [-8]

= 7/16

Therefore, the limit of the given expression, lim[x→81] [(x/(1+sqrt(x)) - 81/10) / (x-81)], is 7/16.

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