Evaluate the Limit

Problem

(lim_x→121)((√(,x)−11)/(x−121))

Solution

1. Identify the indeterminate form by substituting x=121 into the expression, which results in 0/0

2. Factor the denominator using the difference of squares formula, a2−b2=(a−b)*(a+b) where x=(√(,x))2 and 121=11

x−121=(√(,x)−11)*(√(,x)+11)

3. Substitute the factored form into the limit expression.

(lim_x→121)((√(,x)−11)/((√(,x)−11)*(√(,x)+11)))

4. Simplify the expression by canceling the common factor (√(,x)−11) from the numerator and the denominator.

(lim_x→121)(1/(√(,x)+11))

5. Evaluate the limit by substituting x=121 into the simplified expression.

1/(√(,121)+11)=1/(11+11)

6. Calculate the final numerical value.

1/22

Final Answer

(lim_x→121)((√(,x)−11)/(x−121))=1/22

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