Evaluate the Limit

Problem

(lim_x→16)((4−√(,x))/(x−16))

Solution

1. Identify the indeterminate form by substituting x=16 into the expression, which results in (4−√(,16))/(16−16)=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is 4+√(,x)

(lim_x→16)((4−√(,x))/(x−16))⋅(4+√(,x))/(4+√(,x))

3. Expand the numerator using the difference of squares formula (a−b)*(a+b)=a2−b2

(lim_x→16)((16−x)/((x−16)*(4+√(,x))))

4. Simplify the expression by noting that 16 - x = -(x - 16),a*l*l*o*w*i*n*g*ƒ*o*r*t*h*e*c*a*n*c*e*l*l*a*t*i*o*n*o*ƒ*t*h*e*c*o*m*m*o*n*ƒ*a*c*t*o*rx - 16)$ in the numerator and denominator.

(lim_x→16)((−(x−16))/((x−16)*(4+√(,x))))

(lim_x→16)((−1)/(4+√(,x)))

5. Evaluate the limit by substituting x=16 into the simplified expression.

(−1)/(4+√(,16))

(−1)/(4+4)

Final Answer

(lim_x→16)((4−√(,x))/(x−16))=−1/8

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