Evaluate the Limit

Problem

(lim_x→8)((√(3,x)−2)/(x−8))

Solution

1. Identify the indeterminate form by substituting x=8 into the expression, which results in (√(3,8)−2)/(8−8)=0/0

2. Recognize that the denominator x−8 can be treated as a difference of cubes where x=(√(3,x))^3 and 8=2^3

3. Apply the formula for the difference of cubes, a^3−b^3=(a−b)*(a^2+a*b+b^2) to the denominator.

4. Substitute a=√(3,x) and b=2 into the formula to rewrite the denominator.

x−8=(√(3,x)−2)*((√(3,x))^2+2√(3,x)+4)

5. Simplify the original limit expression by canceling the common factor (√(3,x)−2) from the numerator and the denominator.

(√(3,x)−2)/((√(3,x)−2)*(√(3,x)^2+2√(3,x)+4))=1/(√(3,x)^2+2√(3,x)+4)

6. Evaluate the limit by substituting x=8 into the simplified expression.

1/(√(3,8)^2+2√(3,8)+4)=1/(2^2+2*(2)+4)

7. Calculate the final numerical value.

1/(4+4+4)=1/12

Final Answer

(lim_x→8)((√(3,x)−2)/(x−8))=1/12

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