Evaluate the Limit

Problem

(lim_x→1)(((6*x−3)2−9)/(3*x−3))

Solution

1. Identify the indeterminate form by substituting x=1 into the expression, which results in ((6*(1)−3)2−9)/(3*(1)−3)=0/0

2. Expand the squared binomial in the numerator using the identity (a−b)2=a2−2*a*b+b2

(6*x−3)2=36*x2−36*x+9

3. Simplify the numerator by subtracting 9

36*x2−36*x+9−9=36*x2−36*x

4. Factor the numerator and the denominator to isolate the common term.

36*x2−36*x=36*x*(x−1)

3*x−3=3*(x−1)

5. Cancel the common factor (x−1) from the numerator and the denominator, provided x≠1

(36*x*(x−1))/(3*(x−1))=(36*x)/3=12*x

6. Evaluate the limit by substituting x=1 into the simplified expression.

(lim_x→1)(12)*x=12*(1)=12

Final Answer

(lim_x→1)(((6*x−3)2−9)/(3*x−3))=12

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