Find the following limits or state that they do not exist. Ass...

Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→4 x^2 − 16 / 4 − x

Accepted Answer

Welcome back, everyone determine the limit or state that it does not exist limit of T squared minus 25 divided by five minus T. When T approaches five A says negative 10 B says 10 C says zero and D says the limit does not exist. So let's evaluate this limit by trying direct substitution. T approaches five, we are evaluating a rational expression T squared minus 25 divided by five minus T. If we apply direct substitution, we end up with five squared minus 25 which gives us 05 minus five. This gives us zero as well. This is an indeterminate form, meaning we want to simplify the rational expression. And in this case, we can easily factor it out. Let's begin once again with the given limit. When T approaches five, we can see that in the numerator, we have a difference of squares T squared minus five squared because 25 is five squared right? In the denominator that's five minus T, let's apply the factorization formula. Whenever we have a difference of squares A squared minus B squared, we can factor it out as A minus B multiplied by A plus B So that'd be T minus five multiplied by T plus five. That's how we factor out the numerator. And now for the denominator, well, it looks like we have five minus T and we could cancel out T minus five. So what we can do is just factor out the negative sign and then we end up with T minus five. This is really helpful because now we can cancel out the denominator and we end up with lame when T approaches five of negative because they're still a negative sign in the denominator of T plus five. And now we can once again try direct substitution, we end up with negative five plus five. Now five plus five gives us 10. So we end up with negative 10 as our final answer which is option A. Thank you for watching.

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→4 x^2 − 16 / 4 − x

Key Concepts

Limits

Factoring

Indeterminate Forms

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