Video: Finding the One-Sided Limit of a Function from Its Graph at a Point If the Limit Exists

Determine lim_(π‘₯ β†’ βˆ’5⁺) 𝑓(π‘₯).

01:49

Video Transcript

Determine the limit as π‘₯ tends to negative five from above of 𝑓 of π‘₯.

We are given a graph of the function 𝑓 of π‘₯. And we can see that at negative five, that seems to be an asymptote. And therefore, the function 𝑓 of π‘₯ is undefined at π‘₯ equals negative five, but we’re looking for the limit as π‘₯ tends to negative five from above. The little plus sign tells us that we’re only considering values of π‘₯, which are greater than negative five. We’re getting closer and closer to negative five from above.

Looking back to the graph, we can see that 𝑓 of negative four is negative one. 𝑓 of negative 4.5 appears to be negative two. And as π‘₯ gets closer and closer to negative five, always remaining however greater than negative five, we can see that the value of 𝑓 of π‘₯ decreases.

And as there is an asymptote there, in fact it will continue to decrease without bound. So it will go past negative a million, then negative a billion, negative a trillion. Every real number will be passed.

So the limit as π‘₯ tends to negative five from above of 𝑓 of π‘₯ can’t be any real number because as π‘₯ gets closer to negative five, eventually the value of 𝑓 of π‘₯ will become smaller than any number you could name.

Another way of saying this, a shorthand for this is that the limit as π‘₯ tends to negative five from above of 𝑓 of π‘₯ is equal to negative infinity. This doesn’t mean that negative infinity is being considered as a number; this is just a shorthand for the statement we talked about before, which is that as π‘₯ gets closer and closer to negative five from above, 𝑓 of π‘₯ decreases without bound.

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