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

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


2m read
·Nov 11, 2024

So let's see if we can find the limit as x approaches negative one of six x squared plus five x minus one.

Now, the first thing that might jump out at you is this right over here. This expression could be used to define the graph of a parabola. When you think about this, I'm not doing a rigorous proof here; a parabola would look something like this.

This would be an upward opening parabola. It looks something like this; this graph visually is continuous. You don't see any jumps or gaps in it. In general, a part of a quadratic like this is going to be defined for all values of x, for all real numbers, and it's going to be continuous for all real numbers.

So, something that is continuous for all real numbers—well then, the limit as x approaches some real number is going to be the same thing as just evaluating the expression at that real number. So what am I saying? I'm just going to say it another way: We know that some function is continuous at some x value, at x equals a, if and only if—that is, if or if if and only if—the limit as x approaches a of f of x is equal to f of a.

So, I didn't do a rigorous proof here, but just it's conceptually not a big jump to say, okay, well this is just a standard quadratic right over here. It's defined for all real numbers and, in fact, it's continuous for all real numbers.

So we know that this expression could define a continuous function, so that means that the limit as x approaches a for this expression is just the same thing as evaluating this expression at a. In this case, our a is negative 1.

So all I have to do is evaluate this at negative 1. This is going to be 6 times negative 1 squared plus 5 times negative 1 minus one. So that's just one. This is negative five. So it's six minus five minus one, which is equal to zero, and we are done.

More Articles

View All
Welcome to high school biology | High school biology | Khan Academy
It’s very easy to take for granted some of the very amazing things that surround us. In particular, the notion of life on a day-to-day basis — we’re used to it. We are living things; we see living things all around us, and we get preoccupied with other th…
Watch Thousands of Dogs Run Free in This Magical Sanctuary | Short Film Showcase
It was also he was the first dog who got me thinking. So what if I didn’t want to keep? Also, he wasn’t gonna find a woman, anybody else, and he’d be put down. So a hundred dogs became 200 dogs, and then 300 dogs. And then all of a sudden we thought, “Oka…
Constitutional powers of the president | American civics | US government and civics | Khan Academy
So this is here, um, I’m here with Jeffrey Rosen, head of the National Constitution Center, and we’re going to talk about Article Two of the United States Constitution. So, Jeffrey, what does Article Two deal with? It deals with the executive power, the…
Deficits and debt | AP Macroeconomics | Khan Academy
Two terms that you’ve likely heard in the context of government spending, budgets, and borrowing are the terms deficit and debt. They can get a little bit confusing because they’re associated with borrowing in budgets and spending, and they both start wit…
Ridiculously Easy DIY Light Strips! (no soldering)
I want to change my bathroom from this to this. The problem is I want it to not cost a lot, be high quality, and be easy. I mean, is that even possible? Well, after trying out many different options and almost failing multiple times, I finally found a gre…
Wabi-Sabi | A Japanese Philosophy of Perfect Imperfection
The pursuit of perfection has become the norm in today’s world, where chronic dissatisfaction, burnout, depression, and anxiety reign supreme. We’ve subjected ourselves to unrealistic standards and rigorously chase an ideal that’s impossible to reach. Adv…