Determine if Continuous f(x)=(x^2-4)/(x+2)

Problem

ƒ(x)=(x2−4)/(x+2)

Solution

1. Identify the domain of the function by finding where the denominator is zero.

x+2=0

x=−2

2. Determine the continuity at the excluded value. Since the function is a rational function, it is continuous at all points in its domain. However, at x=−2 the function is undefined.

ƒ*(−2)=((−2)2−4)/(−2+2)

ƒ*(−2)=0/0

3. Simplify the expression to check for a removable discontinuity.

ƒ(x)=((x−2)*(x+2))/(x+2)

ƒ(x)=x−2,x≠−2

4. Evaluate the limit as x approaches −2

(lim_x→−2)((x2−4)/(x+2))=(lim_x→−2)(x−2)

(lim_x→−2)(x−2)=−4

5. Conclude that because ƒ*(−2) is undefined, the function is not continuous at x=−2 It is continuous on the intervals (−∞,−2) and (−2,∞)

Final Answer

ƒ(x)=Continuous on *(−∞,−2)∪(−2,∞)

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