Determine if Continuous f(x)=( square root of x-2)/(x^2-9)

Problem

ƒ(x)=√(,x−2)/(x2−9)

Solution

1. Identify the domain of the numerator. The square root function √(,x−2) is defined only when the radicand is non-negative.

x−2≥0

x≥2

2. Identify the values that make the denominator zero. The function is undefined where x2−9=0

x2−9=0

(x−3)*(x+3)=0

x=3,x=−3

3. Determine the intersection of these conditions. The function is defined for all x in the interval [2,∞) except for the values that make the denominator zero. Since x=−3 is already outside the interval [2,∞) we only need to exclude x=3

D=[2,3)∪(3,∞)

4. Apply the definition of continuity. A rational function involving roots is continuous at every point in its domain. Therefore, ƒ(x) is continuous on the intervals [2,3) and (3,∞) It is discontinuous at x=3 (infinite discontinuity) and undefined for x<2

Final Answer

ƒ(x)* is continuous on *[2,3)∪(3,∞)

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