Evaluate the Limit

Problem

(lim_x→144)((√(,x)−12)/(x−144))

Solution

1. Identify the indeterminate form by substituting x=144 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 144=12

x−144=(√(,x)−12)*(√(,x)+12)

3. Rewrite the limit expression using the factored denominator.

(lim_x→144)((√(,x)−12)/((√(,x)−12)*(√(,x)+12)))

4. Simplify the expression by canceling the common factor (√(,x)−12) from the numerator and the denominator.

(lim_x→144)(1/(√(,x)+12))

5. Evaluate the limit by substituting x=144 into the simplified expression.

1/(√(,144)+12)

6. Calculate the final numerical value.

1/(12+12)=1/24

Final Answer

(lim_x→144)((√(,x)−12)/(x−144))=1/24

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