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→3 x − 3 /|x − 3|

Accepted Answer

Welcome back, everyone find the limit or state that it does not exist limit of X minus seven divided by the absolute value of X minus seven. When X approaches seven, there are four answer choices. A one B negative one C zero and D, the limit does not exist before we begin. Let's remember the definition of the absolute value as a function. So if we have the absolute value of X minus seven, it is basically a piecewise function. It is going to be equal to X minus seven. As long as X minus seven is greater than, or equal to zero or negative X minus seven. If X minus seven is less than zero, specifically, if we look at each inequality, we can state that either X is greater than, or equal to seven, we get that by adding seven to both sides. And then for the second inequality, once again, we can add seven to both sides and see that X would be less than seven. And this is exactly what we have in the limit. So we have two possibilities for our limit. The first one is limit when X approaches seven from the right So that'll be X approaching seven from the right. And if X is less than seven, this essentially means that we're looking at X approaching seven from the left. So we have to valley two limits. One of them is limit. When X approaches seven from the right. In the numerator, we have X minus seven in the denominator based on the definition of the piecewise function, we are going to have X minus seven as well. And this gives us a limit of one when X approaches seven from the right, which is basically one, we have a constant function. Now let's look at the limit from the left, we get X minus seven in the numerator. And in the denominator, we have negative X minus seven, we can cancel out the common factor which is X minus seven and we end up with that negative sign. So essentially, we are evaluating the limit when X approaches seven from the left of one divided by negative one, which is negative one and that's negative one. Let's notice that the limit from the left is not equal to the limit from the right of that same point. So we get limit when X approaches seven from the right of F of X, let's call the overall function F of X is not equal to the limit. When X approaches seven from the left of F of X, let's recall that whenever those limits are not equal to each other, this means that the limit at the point itself which is seven does not exist, right? So because the two limits are not equal to each other, the limit at the point itself does not exist. Meaning the answer to this problem is option D, the limit does not exist. Thanks 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→3 x − 3 /|x − 3|

Key Concepts

Limits

Absolute Value Function

One-Sided Limits

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