Determine whether the following statements are true and give a...

Question

Determine whether the following statements are true and give an explanation or counterexample.

a. The value of $\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$ does not exist.

Accepted Answer

Welcome back, everyone determine the validity of the given statement that the value of the limit of X squared minus 25 divided by X minus five as X approaches five does not exist. A says it's true. B says it's false. Now, how can we tell if this statement is true or not? Well, it's true if we can prove that the limit does not exist, but it will be false if we can show that the limit will exist or the limit does exist. OK. So let's try and test to see if we can actually find that limit. So the limit as X approaches five of X squared minus 25 divided by X minus five. No, at the start initially, it might seem like the limit does not exist because if we substitute five into our expression five minus five equals zero. And thus our term will be undefined. However, OK, we could rewrite X squared minus 25 as X minus five multiplied. Let me write that properly here. X minus five multiplied by X plus five because it is the difference of two squares when we write it like that, no, X minus five cancels X minus five and we are left with the limit as X approaches five of X plus five. If we substitute five into that expression, it becomes five plus five, which equals 10. Thus, OK. Thus, the limit exists. OK? The limit exists. And that means this, this statement must be false. B is the correct answer. Thanks for watching everyone. I hope this video helped.

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Determine whether the following statements are true and give an explanation or counterexample.

a. The value of $\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$ does not exist.