Evaluate the Limit

Problem

(lim_x→1)(((10*x−5)2−25)/(5*x−5))

Solution

1. Identify the indeterminate form by substituting x=1 into the expression.

((10*(1)−5)2−25)/(5*(1)−5)=(5−25)/0=0/0

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

(10*x−5)2=100*x2−100*x+25

3. Substitute the expanded form back into the numerator and simplify by combining like terms.

100*x2−100*x+25−25=100*x2−100*x

4. Factor the numerator and the denominator to isolate the common factor (x−1)

100*x2−100*x=100*x*(x−1)

5*x−5=5*(x−1)

5. Simplify the fraction by canceling the common factor (x−1) from the numerator and denominator.

(100*x*(x−1))/(5*(x−1))=(100*x)/5

(100*x)/5=20*x

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

(lim_x→1)(20)*x=20*(1)=20

Final Answer

(lim_x→1)(((10*x−5)2−25)/(5*x−5))=20

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