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### Course: Differential Calculus>Unit 1

Lesson 3: Estimating limits from tables

# One-sided limits from tables

In this video, we explore one-sided limits using tables, focusing on estimating the value a function approaches from either the left or right. We emphasize the importance of understanding the negative and positive superscripts, which indicate the direction we're approaching the limit from.

## Want to join the conversation?

• When can you tell if the limit of a one-sided function does not exist? I am having trouble figuring that out.
• A limit is defined as the value of a function f(x) as x approaches some c value from both sides of said c value. A one-sided limit is the same as a regular limit, but it only requires one side of the function to be approaching the c value. One-sided limits may not exist in the following cases:
-The function goes to infinity (a vertical tangent) at the c value from that side
-The function doesn't approach the c value from that side

The one sided limit still exists in the following cases:
-The limit from the other side is different
-The actual value of the function at the point c doesn't exist (hole), but the one side of the function still approaches a value
-The actual value of the function jumps at the point c, but the one side of the function still approaches a value
-A discontinuity in the function that doesn't affect the ability to determine the limit

Hope this helps!
• Will we learn how to find out the actual limit of a function later in calculus, or is that not possible?
• You will learn how to find the actual limit. It's later on in this unit.
• Wouldn't it be more reasonable to assume that lim(x->-2^-) f(x)=-2500?
Because -20*5=-100, -100*5=-500, -500*5=-2500
• The thing is, why did you stop there? Perhaps -2500 is the value at -2.0001. Maybe there's another value at -2.00001, say -2500 * 5 = -12500. Is that the limit? Or will it multiply by five again if we try a closer number? Just how close do we need to get before we're certain?

If the values started to slow down at some point, then it might be a reasonable estimate. For example:
-2.0001 = 2499
-2.00001 = 2499.5
-2.000001 = 2499.75
In this case, yes, -2500 is probably a good guess. But working just from the table we're given, the numbers appear to simply be increasing exponentially as we approach -2, and for that reason we can assume it probably is unbounded.
• What I must do to be 100% sure about the limit of a function?What is the method for finding the limit of a function with absolute certainty?
• This is a good question. Using tables only suggests what the limit might be, but does not give the limit with certainty. However, limits (if they exist) can be found with certainty by using algebraic techniques such as direct substitution, factoring & reducing, multiplying top and bottom by a conjugate, using trigonometric identities, and using L’Hopital’s rule.
• How to know that the function is unbounded ?
• consider x approaching -2 and y=f(x). We see that as the function grows closer to -2 (-2.1,-2.01,-2.001, -2.00000001, etc...) f(x) falls exponentially (-100, -500, etc..). Plot this on a graph and you'll notice the function is unbounded at x= -2 and that the limit approaching from the left is infinity (or doesn't exist).
• for the first example can the estimate be 1 or 1.5?*
• Not quite. See that as the function approaches x = 1 from the left, f(x) seems to approach 2. Now, with more data, it could be possible that it approaches some other number, but just with the data we have, the limit being 2 is a more reasonable estimate.
• On the second example on this timestamp: 5.00 confused me. Is it unbounded because because f(-2.002) (which is -500) is larger than f(2) (which is -4)? Or I am missing something?
• You might be thinking about this the wrong way.

The idea is that, as you get closer and closer to -2 without actually getting there, it seems like the function is becoming exponentially larger. Notice how in the table, we can get really close, and the trend seems to stick.

Essentially, don't connect the dots on the table, try to sniff out patterns.

Hope this helps.
• Wait, I'm confused, why was the limit to the left negative 4? Shouldn't it be it doesn't exist, since -4 was an f(x) value?
• I don't think the actual value of the function at the specified point is needed in order to determine the limit. You only need values that get closer and closer to that point. Therefore, it is redundant that -4 is a value of f(x) as it does not affect the limit.