How to Find Any Limit: Part 2 (NancyPi)

MIT grad shows how to find the limit at a finite value with a square root, (sin x)/x, or absolute value. To skip ahead: 5) for a SQUARE ROOT in the numerator or denominator (to RATIONALIZE by multiplying by the "CONJUGATE"), skip to time 1:14. 6) for a limit with something of the form (SIN X)/X, skip to time 5:38. 7) for an ABSOLUTE VALUE in the limit expression, skip to time 14:45. Nancy formerly of MathBFF explains the steps.

5) For a SQUARE ROOT in the numerator or denominator: If you try plugging in the value for x and get a 0 in the denominator, and you cannot factor, get a common denominator, or expand to simplify the expression, then if there's a square root in the numerator or denominator, you can try MULTIPLYING by the CONJUGATE. For instance, if you have sqrt(x+1) - 3 in the numerator, you would multiply both the numerator and denominator by sqrt(x+1) + 3 because the "conjugate" just means a two-term expression with the sign flipped in front of the second term. This is a trick or technique that helps simplify because when you multiply out, or FOIL, the numerator you will get terms that cancel. It is best to leave the denominator factored, rather than multiplying out the terms since a factor is likely to cancel. Once you simplify by multiplying on top, combining like terms, and canceling any factors from the top and bottom, try plugging in the value again for x to get an actual limit value.

6) For the form (SIN X)/X in a limit expression: If you try plugging in the value that x is approaches, and you get 0 in the denominator, if your limit expression is something of the form (sin x) over x, there is a trig property that you can use to simplify. The property is that the limit of (sin x)/x, as x approaches 0, is equal to 1. If your expression isn't exactly (sin x)/x but instead has something like 2x or 3x inside the sin function, like sin(2x) over (4x), you can use the same property but first have to rearrange the expression in a way that matches what you need, as shown in the video. NOTE: Be careful not to confuse this trig property with another, very similar, (sin x)/x expression for when x is approaching infinity. That property states that the limit of (sin x)/x, as x approaches infinity, is equal to 0. Check out the video on limits at infinity for an explanation of how to use that expression.

7) For an ABSOLUTE VALUE in your limit expression: If you try plugging in the value for x and get 0 in the denominator, and you have an absolute value in your limit expression, you will probably need to re-write the limit expression using the piecewise definition of the absolute value function. You will then have an expression for the left-side limit and one for the right-side limit. If you evaluate the left side and right side, and the numbers agree, then that is your limit value. If the two sides do not have limit values that agree, then the limit does not exist.

[alexaciarletta]

a literal queen she explains it so simply

[ejembiados]

You've made calculus looks like a child's play. Thanks for this amazing teaching approach

[sneakyninja]

You are saving my life in calculus right now!

[MuhammadAsaad]

Damn if I had known this channel at the start of the year rather than 2 days before exam I could've been a master of Math 😭
I wish our teachers explained things like this this is just superb I never thought these complex math things can ever be simple

[armyguy]

I owe you big time! I've gone back to school to finish my math major and go to dental school and I'm having to relearn all of my calculus (its been 8 years since I first took Calculus 1!). You are helping me so much! I work two jobs full time around school and never get a chance to go see my professors during office hours, but thankfully youtube is never closed! Thank you for your awesome contribution to the learning community =)

[aminesf]

2024 still watching this and enjoying it like it was the first time :) thank you infinitely Nancy :)

[brcarter]

"You're going to be like a but for finding limits at a finite value..."

[brifish]

Can I just thank you for taking the time to edit in the fast-forwards? I'm learning it either way, but it makes it even more classy. Stay classy, Nancy Pi.

[alizahour]

She is amazing. And she teaches very calmly, not like other youtubers who are screaming.

[elsf]

These videos are just so relaxing. I watch even when I know the content :)

[konormccracken]

lmao she has a weird knack of uploading these videos at the most convenient times

[FaithMancuso]

you literally saved my life and taught this in the most simplest form and explained every question I had. thanks :)

[MattAkiyama]

I want to say thank you for these videos on limits. It greatly helped me understand how they work and made the final exam an absolute breeze! Your videos are very understandable.

[en]

I love how you organize everything into blocks like this. It can be overwhelming having the different techniques individually floating around my mind and you make it so much easier :')

[saratoborowski]

Nancy has a talent of making things simple to understand and remember! thank you!

[Gravy]

Currently using these as a refresher for my Cal A midterm. Thank you a lot!

[phillipyovino]

Havent taken a math class in 8 years so my school has me start back at calculus. These videos are amazing. You cover the material well and the way you separate the material is very helpful. Thank you!

[Upfrontgameplay]

Thanks again Nancy, I’m a limit ninja now due to your help! You’re the best!

[samheinrich]

*Love the videos! DON'T GIVE UP NANCY*

[JCAvenilla]

You're a better teacher compared to my college professor 😭😍