Let g(x)={[(x-3)/(sqrt(x+13)-4)," for "x = -13","x!=3],[k," for "x=3]:} g is continuous for all x -13. What is the value of k ? Choose 1 answer: (A) 4 (B) 0 (C) 8 (D) 3

Question

Let  g(x)={[(x-3)/(sqrt(x+13)-4)," for "x >= -13","x!=3],[k," for "x=3]:}  g is continuous for all  x > -13. What is the value of  k ? Choose 1 answer: (A) 4 (B) 0 (C) 8 (D) 3

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Define Function: To ensure the function g(x) is continuous at x = 3, the limit of g(x) as x approaches 3 must equal the value of g(x) at x = 3. We need to find the limit of the first part of the function as x approaches 3.Simplify Expression: We have the function g(x) for x ≠ 3 as: g(x) = (x - 3) / (sqrt(x + 13) - 4) To find the limit as x approaches 3, we can try to simplify the expression to remove the indeterminate form that arises from direct substitution.Multiply by Conjugate: To simplify the expression, we can multiply the numerator and the denominator by the conjugate of the denominator: g(x) = [(x - 3) / (sqrt(x + 13) - 4)] * [(sqrt(x + 13) + 4) / (sqrt(x + 13) + 4)] This will help us get rid of the square root in the denominator.Further Simplify: After multiplying, we get: g(x) = ((x - 3) * (sqrt(x + 13) + 4)) / ((sqrt(x + 13))^2 - 4^2) This simplifies to: g(x) = ((x - 3) * (sqrt(x + 13) + 4)) / (x + 13 - 16)Cancel Terms: Further simplifying the denominator, we get: g(x) = ((x - 3) * (sqrt(x + 13) + 4)) / (x - 3) Now, we can cancel out the (x - 3) terms in the numerator and the denominator, as long as x ≠ 3.Direct Substitution: After canceling out the (x - 3) terms, we are left with: g(x) = sqrt(x + 13) + 4 Now, we can find the limit as x approaches 3 by direct substitution since the expression is no longer indeterminate.Substitute x=3: Substituting x = 3 into the simplified expression, we get: limit as x approaches 3 of g(x) = sqrt(3 + 13) + 4 This simplifies to: limit as x approaches 3 of g(x) = sqrt(16) + 4Simplify Limit: Since sqrt(16) is 4, we have: limit as x approaches 3 of g(x) = 4 + 4 This gives us: limit as x approaches 3 of g(x) = 8Finalize Value: For g(x) to be continuous at x = 3, the value of k must be equal to the limit as x approaches 3 of g(x). Therefore, k = 8.

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