# Introduction to Limits (NancyPi)

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Published on August 31, 2018 by

For HOW TO FIND THE LIMIT (at a finite value), jump to https://youtu.be/nJZm-zp639s
For HOW TO FIND THE LIMIT AT INFINITY, jump to https://youtu.be/nViVR1rImUE

Follow Nancy on Instagram: https://instagram.com/nancypi

1) LIMIT NOTATION and WHAT A LIMIT MEANS: You can read the limit notation as "the limit, as x approaches 1, of f(x)". This means "when x gets very close to 1, what number is y getting very close to?" The limit is always equal to a y-value. It is a way of predicting what y-value we would expect to have, if we tend toward a specific x-value. Why do we need the limit? One reason is that there are sometimes "blindspots" such as gaps (holes) in a function in which we cannot see what the function is doing exactly at a point, but we can see what it is doing as we head toward that point.

2) HOW TO LOOK AT THE GRAPH to find the limit: a) For a removable discontinuity (hole), b) For a removable discontinuity with a point defined above, and c) For a normal line. When you're finding an overall limit, the hidden, implied meaning is that YOU MUST CHECK BOTH SIDES OF THE X-VALUE, from the left and from the right. If both sides give you the same limit value, then that value is your overall limit. In our example, to find the limit from the left side, TRACE X VALUES from the left of 1 but headed toward 1 (the actual motion is to the right), and check to SEE WHAT Y-VALUE the function is tending toward. That y-value is the left-hand limit. To find the limit from the right side, trace x values from the right of 1 but headed toward 1 (the actual motion is to the left), and again check to see what Y-VALUE the function is heading toward. That y-value is the right-hand limit. Since the left limit (2) and the right limit (2) are the same in our example, the overall limit answer is 2. If they were not the same, we could not give a limit value (see #3). IMPORTANT TAKEAWAY: For the limit, we DO NOT CARE what is happening EXACTLY AT THE X-VALUE and ONLY CARE what y-values the function is hitting NEAR the x-value, as we get closer and closer to that x. In other words, the limit, as x approaches 1, of f(x) can equal 2, even if (1) = 3 or some other number different from 2, or even if f(1) is not defined or indeterminate.

3) ONE-SIDED LIMITS (RIGHT-SIDED LIMIT and LEFT-SIDED LIMIT) for a jump discontinuity: as you saw in #2, to find the overall limit, you have to check both the left and right limits. Sometimes the left limit and right limit are not the same. If you get a limit question with notation in which the x is approaching a number but with a plus sign or minus sign as a superscript, that is notation for a one-sided limit. The minus sign means the limit from the left, and the plus sign means the limit from the right. IF THE LEFT limit AND RIGHT limit are NOT THE SAME, then the overall limit DOES NOT EXIST (sometimes written as "DNE"). Even if the left and right limits are different, you can still write the left-sided limit and right-sided limit values separately.

4) LIMITS in which X APPROACHES INFINITY (or negative infinity): Another "blindspot" is when x goes toward infinity or negative infinity. Since we can never "see" exactly at infinity (or negative infinity), we can use the idea of the limit to say what y-value it looks like the function is headed toward when our x value approaches infinity. If x is approaching INFINITY, TRACE x values TOWARD THE RIGHT (the large positive direction) on the graph, and see what y-value the function is approaching. That y-value is the limit. Note that the function may be approaching an asymptote. If x is approaching NEGATIVE INFINITY, trace x values TOWARD THE LEFT (the large negative direction), and check what y-value the function is getting closer and closer to on the graph. That y-value is the limit.

For more of my calculus and precalculus math videos, check out: http://nancypi.com

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• Crazy, what she says is as good as how she looks. Thanks for giving a great explanation for us.

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• I really liked the description below the video. You explained the whole thing there also.

• Okay, having a difficult time focusing but that help me understand it a bit more for how it using the word Approach instead of Equal. Just like how Y could never reach asymptote, but as X goes on, it gonna get really close to the asymptote. And that also make sense for why we dont care about the exact point, but the point around it.

• Damn! I am just focusing on her.

• Thank you a lot!

• Not to confuse things, but….the third example made sense EXCEPT for the fact that functions CAN cross horizontal and oblique asymptotes. Only vertical asymptotes are domain restrictions (cannon be crossed since the function is undefined at that domain value). Horizontal and oblique asymptotes are more concerned with the end behavior of the functions as they approach infinitely (both positive and negative)…SO, how can you determine the limit at infinity when you can see out that far so you can’t know for certain that the function will never cross it? Yes, it approaches 0 and -1 near (x=0), but how can we make that assumption? (We CAN make that assumption with vertical limits because the function will always approach but never cross it as I stated earlier). Just curious! Great video and thanks for the fundamental explanations before jumping right into solving problems. Much appreciated and I look forward to watching more of your videos!

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• hi dear i have a confusion

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• my calc teacher this week spoke too fast in a thick accent while teaching us limits so I couldn’t understand or even pay attention, so after class I find this video and now I’m too distracted by how attractive Nancy is… I’m fucked

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• so, your explanation of limits/equal signs right around 5:00 kinda backs up an argument i had on another thread. the limit of something is _not_ really equal to that thing, but is what that thing _approaches_ . without referencing euler’s notes, you back me up quite nicely, thanks!

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mummy boli, “ye kya hai , maine kaha calculus ,
mom had some doubts “.

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• How do we know the limit is 2, and not say 2.00000001?

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• 1:27

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What do you do?

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• Would it be just as easy to say that when f(x) = 2 that the limit defines <2 and >2 which would be the actual definition of the limit which is why itself is undefined? This would be because it shows a transition. A basic example might be f(x) = 9.8 m/s/s. It would show f = ma between an (x) value of t in seconds. It would give force 2 ranges in the domain of seconds. it’s like if the speed limit is 45. Below 45 is safe, above 45 is not safe, can 45 be both safe and unsafe? 🙂

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