Evaluate the Limit

Problem

(lim_x→2)((x2+x−6)/(2−x))

Solution

1. Substitute the value x=2 into the expression to check for an indeterminate form.

(2+2−6)/(2−2)=0/0

2. Factor the numerator x2+x−6 by finding two numbers that multiply to −6 and add to 1

x2+x−6=(x+3)*(x−2)

3. Rewrite the denominator 2 - xb*y*ƒ*a*c*t*o*r*i*n*g*o*u*t*a1$ to make it match a factor in the numerator.

2−x=−(x−2)

4. Substitute the factored forms back into the limit expression.

(lim_x→2)(((x+3)*(x−2))/(−(x−2)))

5. Cancel the common factor (x−2) from the numerator and the denominator.

(lim_x→2)((x+3)/(−1))

6. Evaluate the limit by substituting x=2 into the simplified expression.

(2+3)/(−1)=−5

Final Answer

(lim_x→2)((x2+x−6)/(2−x))=−5

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