Evaluate the Limit

Problem

(lim_x→49)((√(,x)−7)/(x−49))

Solution

1. Identify the indeterminate form by substituting x=49 into the expression, which results in 0/0

2. Factor the denominator using the difference of squares formula, a^2−b^2=(a−b)*(a+b) where x=(√(,x))^2 and 49=7^2

3. Rewrite the expression with the factored denominator.

(√(,x)−7)/((√(,x)−7)*(√(,x)+7))

4. Simplify the expression by canceling the common factor (√(,x)−7) from the numerator and the denominator.

1/(√(,x)+7)

5. Evaluate the limit by substituting x=49 into the simplified expression.

1/(√(,49)+7)

6. Calculate the final numerical value.

1/(7+7)=1/14

Final Answer

(lim_x→49)((√(,x)−7)/(x−49))=1/14

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