Evaluate the limit: lim_(x rarr12)(sqrt(x+4)-4)/(x-12)

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Evaluate the limit:  lim_(x rarr12)(sqrt(x+4)-4)/(x-12)

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Identify Indeterminate Form: Identify the indeterminate form. We need to evaluate the limit of the function as x approaches 12. Let's first plug in the value of x into the function to see if it results in an indeterminate form. lim_(x -> 12)(sqrt(x+4)-4)/(x-12) = (sqrt(12+4)-4)/(12-12) = (sqrt(16)-4)/0 = (4-4)/0 = 0/0 This is an indeterminate form, so we cannot directly calculate the limit by substitution.Algebraic Manipulation: Apply algebraic manipulation to simplify the expression. To resolve the indeterminate form, we can multiply the numerator and the denominator by the conjugate of the numerator. The conjugate of sqrt(x+4)-4 is sqrt(x+4)+4. lim_(x -> 12)((sqrt(x+4)-4)/(x-12)) * ((sqrt(x+4)+4)/(sqrt(x+4)+4)) = lim_(x -> 12)((x+4-16)/(x-12)(sqrt(x+4)+4))Simplify Expression: Simplify the expression further. Now, we simplify the numerator and the denominator. lim_(x -> 12)((x+4-16)/(x-12)(sqrt(x+4)+4)) = lim_(x -> 12)((x-12)/(x-12)(sqrt(x+4)+4)) We can see that (x-12) in the numerator and denominator will cancel out, as long as x is not equal to 12. lim_(x -> 12)(1/(sqrt(x+4)+4))Evaluate Limit: Evaluate the limit of the simplified expression. Now that we have simplified the expression, we can substitute x = 12 directly into the remaining expression. lim_(x -> 12)(1/(sqrt(x+4)+4)) = 1/(sqrt(12+4)+4) = 1/(sqrt(16)+4) = 1/(4+4) = 1/8

Evaluate the limit: $\lim _{x \rightarrow 12} \frac{\sqrt{x+4}-4}{x-12}$

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Evaluate the limit: $\lim _{x \rightarrow 12} \frac{\sqrt{x+4}-4}{x-12}$