yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Undefined limits by direct substitution | Limits and continuity | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

Let's see if we can figure out the limit of x over natural log of x as x approaches one. Like always, pause this video and see if you can figure it out on your own.

Well, we know from our limit properties this is going to be the same thing as the limit as x approaches one of x over the limit, the limit as x approaches one of the natural log of x.

Now, this top limit, the one I have in magenta, this is pretty straightforward. If we had the graph of y equals x, that would be continuous everywhere; it's defined for all real numbers and it's continuous at all real numbers. So, if it's continuous, the limit as x approaches one of x is just going to be this evaluated at x equals one.

So, this is just going to be one. We just put a one in for this x, so the numerator here would just evaluate to a one. Then the denominator, natural log of x, is not defined for all x's and therefore it isn't continuous everywhere. But it is continuous at x = 1.

Since it is continuous at x = 1, then the limit here is just going to be the natural log evaluated at x = 1. So this is just going to be the natural log, the natural log of one, which of course is zero.

e to the 0 power is 1, so this is all going to be equal to, this is going to be equal to, we just evaluate it: 1 over 0.

Now we face a bit of a conundrum. 1/0 is not defined. If it was 0 over 0, we wouldn't necessarily be done yet; that's an indeterminate form. As we will learn in the future, there are tools we can apply when we're trying to find limits and we evaluate it like this and we get 0 over 0.

But 1 over 0, this is undefined, which tells us that this limit does not exist. So, does not exist, and we are done.

More Articles

View All
Symmetry of second partial derivatives
So in the last couple videos I talked about partial derivatives of multivariable functions, and here I want to talk about second partial derivatives. So, I’m going to write some kind of multivariable function. Let’s say it’s, um, I don’t know, sin of x * …
Why It Was Almost Impossible to Make the Blue LED
LEDs don’t get their color from their plastic covers. And you can see that because here is a transparent LED that also glows the same red color. The color of the light comes from the electronics themselves. The casing just helps us tell different LEDs apa…
Lion Falls From Tree During Rescue From a Wire Snare Injury | National Geographic
We pretty much picked up on finding that Orinda at a point where she was treated about two weeks back. So even though the snare was removed, we saw that over about a two and a half week period, her conditions started again to rapidly decline. So we find …
Top 10 Most Expensive Headphones In The World
Today we’re looking at the top 10 most expensive headphones in the world. Welcome to A Lux Calm, the place where future billionaires come to get inspired! Hey there, Alack Sirs, today is a great day to pump up the volume and listen up because we’re talki…
Stock Splits are Secretly Pumping the Stock Market
Stock splits, they’re supposed to be totally irrelevant, right? They don’t change anything about the company, they don’t change anything about the valuation, they don’t change anything about the investing thesis. Well, bizarrely, stock splits are somehow …
Area with fraction division example
We’re told a yoga mat is three-fifths of a meter wide and has an area of one and two twenty-fifths square meters. What is the length of the mat? Well, we know that length times width is going to give you area. Or another way of thinking about it is if th…