Determine if Continuous f(x)=(x^2-16)/(x-4)

Problem

ƒ(x)=(x2−16)/(x−4)

Solution

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

x−4=0

x=4

2. Determine the continuity at the point x=4 Since the function is undefined at this value, ƒ(x) is discontinuous at x=4

3. Simplify the expression to analyze the type of discontinuity.

ƒ(x)=((x−4)*(x+4))/(x−4)

ƒ(x)=x+4

4. Evaluate the limit as x approaches 4.

(lim_x→4)((x2−16)/(x−4))=(lim_x→4)(x+4)

(lim_x→4)(x+4)=8

5. Classify the discontinuity. Because the limit exists but the function value ƒ(4) does not, there is a removable discontinuity at x=4

Final Answer

ƒ(x)=(x2−16)/(x−4)* is discontinuous at *x=4

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