Define g(4) in a way that extends g(x) = (x² − 16)/(x² − 3x − ...

Question

Define g(4) in a way that extends g(x) = (x² − 16)/(x² − 3x − 4) to be continuous at x = 4.

Accepted Answer

Welcome back, everyone. Find the F 5 that makes the function F X equals X2 minus 25 divided by X2 minus 7 X plus 10 continuous at X equals 5. Were given 4 answer choices A F 5 equals -103, B 10/3, C 30s, and D 10. So essentially what we're going to do is check for the continuity of this function. We're going to evaluate f of 5 by using the right substitution, which gives us 52 minus 25 in the numerator and 52 minus 7 multiplied by 5 + 10 in the denominator. And if we try to evaluate it, well, we're going to get an indeterminate form 0 divided by 0, right? We cannot divide it by 0, so our function is not continuous at X equals 5. So we want to make sure that it becomes continuous. We want to identify what value makes it continuous at X equals 5. So we want to use the condition for continuity which says that. Our limit as X approaches 5 of F of X must be equal to the value of the function at 5, right? So what we want to do is simply evaluate that limit. Limit as X approaches 5 of x2 minus 25 divided by X2 minus 7 x + 10, and we can solve for this limit by using a complete factorization of the numerator and the denominator. So the numerator is a difference of squares. X2 minus 52 gives us. X minus 5 multiplied by x + 5 and we also have a quadratic polynomial in the denominator right which can be factored out as x minus 5 multiplied by x minus 2. And the reason why this works is because we are able to cancel out the whole x minus 5 is a common factor. Which allows us to now perform a direct substitution. So we have X + 5 in the numerator, which gives us 5 + 5, and we also have X minus 2 in the denominator. This gives us 5 minus 2. Now, if we evaluate the result, this gives us 10/3, right? So we have evaluated the limit, and to make our function continuous, this value must be equal to f of 5, right? To make sure that our function continuous, this limit must be equal to F of 5 as well. And therefore, the correct answer to this problem corresponds to the answer choice B. Thank you for watching.

Define g(4) in a way that extends g(x) = (x² − 16)/(x² − 3x − 4) to be continuous at x = 4.

Key Concepts

Continuity of Functions

Limits

Rational Functions

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