Evaluate the Limit

Problem

(lim_x→6)((x3−36*x)/(x−6))

Solution

1. Identify the indeterminate form by substituting x=6 into the expression, which results in 0/0

2. Factor out the common term x from the numerator.

(x*(x2−36))/(x−6)

3. Factor the difference of squares x2−36 in the numerator.

(x*(x−6)*(x+6))/(x−6)

4. Simplify the expression by canceling the common factor (x−6) from the numerator and the denominator.

x*(x+6)

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

6*(6+6)

6. Calculate the final numerical value.

6*(12)=72

Final Answer

(lim_x→6)((x3−36*x)/(x−6))=72

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