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Question Video: Discussing the Existence of a Limit Mathematics • Higher Education

Discuss the existence of lim_(π‘₯ β†’ 2) 1/|π‘₯ βˆ’ 2|.

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Video Transcript

Discuss the existence of the limit as π‘₯ approaches two of one over the absolute value of π‘₯ minus two.

In this question, we are given a limit and asked to consider the existence of this limit. We can recall that for a limit to exist, we need the outputs of the function to approach a finite value as the inputs approach the point from either side. This is equivalent to saying that we need the left and right limit of the function at this point to both exist and both be equal.

There are many ways in which we can analyze this limit. For instance, we can sketch the graph of this function by first noting it can be rewritten as the absolute value of one over π‘₯ minus two. We can then sketch a graph of this function by noting that it is a transformation of the graph of the reciprocal function. We translate the graph two units to the right and then reflect any part of the curve below the π‘₯-axis through the π‘₯-axis.

Applying these transformations to the graph of the reciprocal curve gives us the following sketch. We can note that the vertical asymptote is translated two units to the right to the line π‘₯ equals two and the horizontal asymptote is still at 𝑦 equals zero. We can use this sketch to analyze the left and right limits as π‘₯ approaches two for this function. First, we can note that as the values of π‘₯ approach two from the left, the curve is unbounded and approaches positive ∞. We can therefore say that the limit as π‘₯ approaches two from the left of one over the absolute value of π‘₯ minus two is ∞. It is also worth reiterating here that saying a limit is equal to ∞ also means that it does not exist. However, it is useful to note that the limit is ∞.

We have a very similar story if we analyze the graph with values of π‘₯ approaching two from the right. We see that the graph is unbounded towards positive ∞, so the outputs of the function approach positive ∞ when π‘₯ approaches two from the right. So, once again, we can say that the limit as π‘₯ approaches two from the right of this function does not exist. However, we can say that its limit is positive ∞.

Since the left and right limit are both ∞, we can say that the limit is equal to ∞. However, this does mean that the limit does not exist. Hence, the limit does not exist, but the limit as π‘₯ approaches two of one over the absolute value of π‘₯ minus two is equal to ∞.

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