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Great explanation ❤

Although your graph may fly off the handle, no need to jump off the deep end. With these steps, I hope your exponential limits become easier.

Thank you, you have no idea how much this helped me!!!

Don't forget what you already learned about limits. It will come in handy when handling trig limits as well.

Time to squeeze your sandwiches to help us find our limits. 🍔

With these properties, the future of your limit evaluations should be much brighter :)

This guy is AMAZING!!!

Interesting! Dropped you a like :)

To be fair, plugging in values near 0 to show lim_{x->0} sin(x)/x isn't as rigorous since things can change rapidly close to 0. Assuming we don't use calculus, we could try using the squeeze theorem instead to show that result.

Since regular trig functions are mostly continuous, you should be able to take advantage of simply plugging in your x. But for the more complicated trig limits, I hope these shortcuts help out a bit.

With a quick shortcut, you can save loads of time finding these basic limits at infinity.

Shortcuts are always nice, but be sure to understand why they work too.

K.I.S.S. -> That is the name of the game. Simplify where you can to remove those removable discontinuities.

With three different possible outcomes, it is important that you remember that each one represents a different type of continuity status for your function at the point of interest.

By understanding the overall make up of your discontinuity, you do not even need to see the formula to tell if the function has a limit or not.

Limits are all about behavior... Not actual values... Do not be confused :)

With 3 different types of discontinuities, can you identify them all?

Does your infinite geometric series converge or diverge? Gotta start there before trying to use your formula to evaluate it.

Just like our arithmetic series, geometric series follow similar rules. Just make sure you start with the correct sequence type.

Sometimes simply having the formula is not enough. I hope this help you know how to use it.

Start with your sequence and add up all of its elements.. Just make sure it is an arithmetic sequence.

Take your two terms -> Fill in Magic Section -> Poof -> Get your explicit and recursive formula.. Simple right??? :)

Seems like the magic trick is simply "start with the explicit formula" when finding these formulas...

Get your mirror out but this will look a lot like what we did for arithmetic sequences.

With a simple plug and chug method, you should be able to find the terms of your geometric sequence with no problem.

Common ratio or common multiplier... Which one makes more sense to you?

Now it should be super easy to find your remainders.

If you are a fan of the old "Plug and Chug" philosophy, then you should be a fan of the Factor Theorem.

I skipped ahead to the middle of the video and noticed a misconception regarding "complex number OR real number". Technically, all real numbers are considered complex numbers, so that statement may be misleading. However, I understand you likely meant complex numbers with imaginary parts versus those without.

The two formulas do not have to be mutually exclusive. Let them give helping hands to each other.

With just a little bit of information, you can still create your explicit formula.

Whether you are trying to find your first term or your fiftieth term, be sure you are using the formula that best suits your needs.

Is this only for x²?

Ial from india

YourEnglish is good

Were u from

Hi sir

If you are increasing by a fixed amount, that is a definite sign that you are working with an arithmetic sequence.

I hope this quick explanation reaches the one who needs it. As is true of all of math, never be afraid to ask why something is the way it is. Even if it may seem obvious to others.

Don't waste time finding your roots if all you care about is the nature of the roots you have. Use your discriminant.

Complex roots are now able to be found since we are now aware of both imaginary and complex numbers.

With complex conjugates, say goodbye to your imaginary numbers!

Distribute or FOIL... Either method should suffice :)

Group your like terms and add them. Don't forget to distribute your negative if you have to.

Although the name sounds a bit complicated, complex numbers do not have to be too complex :)

😎

Imagine a world where numbers aren't real 😁

To repeat or not to repeat... Is that what makes it rational??

There is more to be continuous than just connecting some dots.

The trick behind the magic is staying on the axis of interest.. Make your other variable = 0.