Let

Question

Let $g\left(x ight)=\frac{x^3-4x}{8\left|x-2 ight|}$ .

Make a conjecture about the values of ${\displaystyle\lim_{x o2^{-}}g\left(x ight)}$ , ${\displaystyle\lim_{x o2^{+}}g\left(x ight)}$ , and ${\displaystyle\lim_{x o2}g\left(x ight)}$ or state that they do not exist.

Accepted Answer

Welcome back everyone. Using the given table of values for F FX equals X cubed minus 25 X divided by 125 multiplied by the absolute value of X minus five. What are the reasonable values of the limit of F FX? As X approaches five from the left, the limit of F of X as X approaches five from the right. And the limit of F FX of as X approaches five A says that as X approaches five from the left, F FX is 2/5 as it approaches five from the right, it's negative 2/5. And as it approaches five, it equals zero, B says they are negative 1/5 1 5th and that the limit does not exist respectively. C says that it's negative 2/5 2 5th and it does not exist. And D says it's negative 2/5 2 5th and 2/5. Now let's use our table to first figure out the value of the limit of F FX as X approaches five from the left. OK. Now here these values at the top, these X values at the top of our table represent X approaching five from the left. OK. And as X approaches far from the left, if we look at the corresponding F FX values notice that those F FX values approach OK, negative 0.4. OK. So in other words, F FX approaches negative 0.4. And if we write that as a fraction, negative 0.4 equals negative to fifths. Therefore, from the first two rows of white table, we can tell that the limit as X approaches P from the left of F FX is equal to negative to fifths. Now, what about our limit from the right? Well, notice here, OK, that our term X is approaching five from the right. OK. In our third row, I know as X approaches five from the right, then if we look at the F of X terms, OK, what do you notice, will you notice that they are getting closer to 0.4? In other words, no F FX approach is 0.4 as X approaches five from the right and we can rewrite 0.4 as 2/5. Therefore, the limit of F FX as X approaches five from the right is going to be equal to 2/5. Now, finally, what is going to be the limit as X approaches five of F FX? Well, we can tell that limit. OK, let me make some space down here. We call that our limit here. The limit of a value as X approaches a value of FX exists. OK. If the limit from the left hand side, X approaches A from the left of F FX is equal to the limit as X approaches A from the right hand side of F FX. So if the left hand limit is equal to the right hand limit, then the limit exists and is equal to that value. Now is that the case from our table? Well notice here that the limit as X approaches five from the left is negative 2/5 while it approaches five from the right is positive 2/5. OK. So in our problem since the limit as X approaches five from the left of F FX is not equal to the limit as X approaches far from the right of F FX, the limit. OK. We can say then that the limit as X approaches five, you know, let me actually write it out the limit as X approaches five of F FX does not exist. OK. So that's what we can conclude because the two limits from either side are not the same. Therefore, when we look on our results, that means C is the correct answer choice. Thanks a lot for watching everyone. I hope this video helped.

Let $g\left(x\right)=\frac{x^3-4x}{8\left|x-2\right|}$ . <IMAGE>
Make a conjecture about the values of ${\displaystyle\lim_{x\to2^{-}}g\left(x\right)}$ , ${\displaystyle\lim_{x\to2^{+}}g\left(x\right)}$ , and ${\displaystyle\lim_{x\to2}g\left(x\right)}$ or state that they do not exist.

Key Concepts

Limits

Piecewise Functions

Continuity

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