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

Question

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.

The limit lim x→a f(x) / g(x) does not exist if g(a)=0.

Accepted Answer

Welcome back, everyone. Assess the validity of the statement. The limit limit as X approaches C of P X divided by Q of X does not exist if Q of C is equal to 0. Assume C is a finite number. Select a correct counterexample among the provided options if the statement is false. And we're given for answer. Choices A says true, B, C, and D those state that the statement is false, giving us different limits. For B it says limit as x approaches -2 of X plus 2 divided by X is 0, for C, it says limit as X approaches 3 of X2 minus 9 divided by X minus 3 is equal to 6, and for D. It says, limit as X approaches 2 of X2 + X minus 6 divided by X minus 3 is equal to 3. So now let's consider this example and let's consider rational functions. Let's suppose that we have a rational function in the form of P X divided by Q X. As a function, we know that rational functions are undefined if the denominator is equal to 0. So the domain of such a function consists of the X values that Do not make the denominator equal to 0, right? However, if we add a limit, this is not the case, right? The limit can exist approaching a point that makes the denominator equal to 0. For example, let's remember the indeterminate forms. If we evaluate a limit, right, and we end up with 0 divided by 0 by direct substitution, this is an indeterminate form, which can still have a finite limit. Another form that we can have is simply a number. Such as a divided by 0, which gives us some sort of infinity. So we can first of all say that the answer to such a problem is false. Yes, rational functions are not defined for denominator of zero, but limit for such a function can exist. Now we're given three possible answers B, C, and D. We can easily exclude B. Because if we look at the X value, it's approaching -2, and this doesn't make the denominator equal to 0. And if we can also exclude option D, X approaches 2 if we subsid 2 into the denominator, while essentially 2 minus 3 gives us -1 and that's not 0. So we end up with the correct answer choice C. Because x approaches 33 minus 3 gives a 0 in the denominator. Now let's analyze that specific limit to prove that we are correct and the given limit exists. If we take a limit as x approaches 3 of X2 minus 9 divided by X minus 3. We first will have to apply the right substitution, and if we substitute X equals 3, we got 9 minus 9 in the numerator, that's 0, and in the denominator 3 minus 3 gives us 0. So that's an indeterminate form. We already discussed that specific form and to eliminate that indeterminate form, we simply have to apply factorization for the numerator because it involves a difference of squares. Now x2 minus 9 can be factored out as x minus 3 multiplied by x + 3, and this is really useful because we have x minus 3 as a factor in the denominator, and those canceled shutter out, and from here we can apply the direct substitution once again. We're left with X + 3, so the answer will be 3 + 3, which gives us a finite number. That's 6, so the limit exists, right, even though our denominator is equal to 0. So we have concluded the final answer, which is option C. Thank you for watching.

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.
The limit lim x→a f(x) / g(x) does not exist if g(a)=0.

Key Concepts

Limits

Indeterminate Forms

Continuity and Discontinuity

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