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![Let
f(x)={[(x+1)/(sqrt(x+5)-2)," for "x >= -5","x!=-1],[k," for "x=-1]:}
f is continuous for all
x > -5.
What is the value of
k ?
Choose 1 answer:
(A) -4
(B) 2
(C) 0
(D) 4](https://externalquestionsimg-a4fuc2b6e0gtdqge.z01.azurefd.net/external-question-images/images/images/ka/APCollegeCalculusAB/Limits%20and%20continuity/Removing%20discontinuities/Q-13.png)
Let is continuous for all .What is the value of ?Choose answer:(A) (B) (C) (D)
Full solution
Q. Let is continuous for all .What is the value of ?Choose answer:(A) (B) (C) (D)
- Define Limit Approach: To find the value of that makes continuous at , we need to ensure that the limit of as approaches from the left is equal to the value of at .
- Find Limit Expression: First, we will find the limit of the function as approaches from the left. This means we will use the piece of the function defined for and , which is the rational expression .
- Algebraic Manipulation: To find the limit as approaches , we can try direct substitution to see if the expression is defined at . Substituting into the expression gives us , which is an indeterminate form. This means we need to use algebraic manipulation to simplify the expression before finding the limit.
- Rationalize Denominator: To eliminate the indeterminate form, we can multiply the numerator and denominator by the conjugate of the denominator. The conjugate of is . This will help us rationalize the denominator.
- Simplify Expression: Multiplying the numerator and denominator by the conjugate, we get:
- Cancel Terms: Simplifying the denominator using the difference of squares, we get:
- Substitute : Further simplifying the denominator, we get:
- Determine Value of : Now we can cancel out the terms in the numerator and denominator, as long as is not equal to (which is not a problem since we are considering the limit as approaches , not the value at ):
- Check Answer Choices: Now we can substitute into the simplified expression to find the limit:
- Check Answer Choices: Now we can substitute into the simplified expression to find the limit: Since is continuous at , the limit as approaches from the left must be equal to the value of at . Therefore, must be equal to .
- Check Answer Choices: Now we can substitute into the simplified expression to find the limit: Since is continuous at , the limit as approaches from the left must be equal to the value of at . Therefore, must be equal to . We can now check the answer choices to see which one matches our result. The correct answer is (D) .
