Let h(x)={[(sqrt(x+26)-5)/(x+1)," for "x = -26","x!=-1],[k," for "x=-1]:} h is continuous for all x -26. What is the value of k ? Choose 1 answer: (A) -(1)/(5) (B) 1 (C) (1)/(10) (D) 0

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Let  h(x)={[(sqrt(x+26)-5)/(x+1)," for "x >= -26","x!=-1],[k," for "x=-1]:}  h is continuous for all  x > -26. What is the value of  k ? Choose 1 answer: (A)  -(1)/(5) (B) 1 (C)  (1)/(10) (D) 0

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Define Limit Approach: To find the value of k that makes the function h(x) continuous at x = -1, we need to ensure that the limit of h(x) as x approaches -1 from the left is equal to the limit of h(x) as x approaches -1 from the right. Since the function is defined differently at x = -1, we need to find the limit of the first part of the function as x approaches -1.Calculate Limit of First Part: We will calculate the limit of the first part of the function as x approaches -1: lim (x -> -1) [(sqrt(x+26)-5)/(x+1)] To do this, we can try direct substitution to see if the limit exists.Resolve Indeterminate Form: Substituting x = -1 into the expression, we get: [(sqrt(-1+26)-5)/(-1+1)] = [(sqrt(25)-5)/(0)] = [(5-5)/(0)] = [0/0] This is an indeterminate form, so we cannot use direct substitution. We need to use another method to find the limit.Rationalize Numerator: To resolve the indeterminate form, we can rationalize the numerator by multiplying the numerator and denominator by the conjugate of the numerator: [(sqrt(x+26)-5)/(x+1)] * [(sqrt(x+26)+5)/(sqrt(x+26)+5)] This will help us simplify the expression and potentially eliminate the indeterminate form.Simplify Expression: Multiplying the expression by the conjugate, we get: [(x+26 - 25)/(x+1)(sqrt(x+26)+5)] = [(x+1)/(x+1)(sqrt(x+26)+5)] We can now simplify the expression by canceling out the (x+1) terms in the numerator and denominator.Evaluate Limit: After canceling out the (x+1) terms, we are left with: 1/(sqrt(x+26)+5) Now we can evaluate the limit as x approaches -1.Substitute x = -1: Substituting x = -1 into the simplified expression, we get: 1/(sqrt(-1+26)+5) = 1/(sqrt(25)+5) = 1/(5+5) = 1/10 So the limit of h(x) as x approaches -1 from the left is 1/10.Find Value of k: Since h(x) is continuous at x = -1, the value of k must be equal to the limit we just found. Therefore, k = 1/10.

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