Evaluate the Limit

Problem

(lim_a→25)((√(,a)−5)/(a−25))

Solution

1. Identify the indeterminate form by substituting a=25 into the expression, which results in 0/0

2. Factor the denominator using the difference of squares formula, x2−y2=(x−y)*(x+y) where a=(√(,a))2 and 25=5

a−25=(√(,a)−5)*(√(,a)+5)

3. Substitute the factored form back into the limit expression.

(lim_a→25)((√(,a)−5)/((√(,a)−5)*(√(,a)+5)))

4. Simplify the expression by canceling the common factor (√(,a)−5) from the numerator and the denominator.

(lim_a→25)(1/(√(,a)+5))

5. Evaluate the limit by substituting a=25 into the simplified expression.

1/(√(,25)+5)=1/(5+5)

1/10

Final Answer

(lim_a→25)((√(,a)−5)/(a−25))=1/10

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