Evaluate the Limit

Problem

(lim_x→225)((√(,x)−15)/(x−225))

Solution

1. Identify the indeterminate form by substituting x=225 into the expression, which results in (15−15)/(225−225)=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is √(,x)+15

(√(,x)−15)/(x−225)⋅(√(,x)+15)/(√(,x)+15)

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

((√(,x))2−15)/((x−225)*(√(,x)+15))

(x−225)/((x−225)*(√(,x)+15))

4. Cancel the common factor of x−225 from the numerator and the denominator, provided x≠225

1/(√(,x)+15)

5. Evaluate the limit by substituting x=225 into the simplified expression.

1/(√(,225)+15)

1/(15+15)

Final Answer

(lim_x→225)((√(,x)−15)/(x−225))=1/30

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