Evaluate the Limit

Problem

(lim_x→100)((√(,x)−10)/(x−100))

Solution

1. Identify the indeterminate form by substituting x=100 into the expression, which results in (√(,100)−10)/(100−100)=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is √(,x)+10

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

((√(,x)−10)*(√(,x)+10))/((x−100)*(√(,x)+10))=(x−100)/((x−100)*(√(,x)+10))

4. Cancel the common factor of x−100 from the numerator and the denominator, provided x≠100

1/(√(,x)+10)

5. Evaluate the limit by substituting x=100 into the simplified expression.

1/(√(,100)+10)=1/(10+10)=1/20

Final Answer

(lim_x→100)((√(,x)−10)/(x−100))=1/20

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