So the tea act and stamp act and stuff was all part of the Townshend acts?

# Homework Help Topic

**Nioma0nwat3r**#146

Yeah they were more than one law. But the whole idea of learning about the townshend acts is usually to show how the British taxed so many ridiculous things without the consent of the people, and that made everyone really mad. After much discussion, parliament took away most of the taxes, but kept the tax on Tea, which lead to the Boston Tea Party.

**ptx.serena**#147

Has anyone here taken calculus, or is taking it? Because if you have I need help lol

**ptx.serena**#153

YAY! Iām in AP calc AB

Have you done finding limits algebraically? My teacher explained it to us quickly, but I didnāt really get itā¦do you have a simple way of explaining it?

**Nioma0nwat3r**#155

Finding limits algebraically? Yeah. Give me a moment to grab my notes and then Iāll be right back with a picture. Cause thereās likeā¦ three different ways to solve algebraically, but they all come back down to one thing.

**Nioma0nwat3r**#157

Alright, so the first step is called āDirect substitutionā. Thatās when you plug the number the limit is going to straight into the equation, which works for equations like x^2, but become problematic for equations that end up as 0/0.

For equations like the above, you have three choices (four really but I couldnāt care less for the fourth. Itās useless and can always be found another way).

Your options are: Factor and cancel, Rationalize and cancel, and common denominator.

Factor and Cancel will work best for this equation. (x^2 - 9) factors into (x + 3) and (x - 3)

Now you can substitute in 3 for x (direct substitution again) and get 6 as your limit.

The basic idea of factor and cancel is literally that: factor either the numerator or denominator, and cancel out any unnecessary expressions.

(End of Part 1. I donāt want to overwhelm you so any questions?)

**ptx.serena**#158

Ohhh ok, thanks! I kept getting 0/0 for almost all of my answers and I knew that couldnāt be right. No questions yet, I think I understand factor and cancel.

**Nioma0nwat3r**#159

Alright, the next one is rationalize and cancel.

Letās take this equation:

As you can see, the answer when you do direct substitution is 0/0, and thatās not what we want.

Now. This is where most of the mistakes are made. What you want to do, is get rid of the square root, so to do that, you must multiply the numeratorās equation by a fractional 1. Numerator and denominator must be the OPPOSITE equation as the numerator. Itās hard to explain, so hereās a picture. (as shown):

This leaves you with a new numerator. Donāt multiply out the denominator.

As you can see, you can now cancel out the (x-4) from both numerator and denominator. After another direct substitution, you get: 1/4

Questions?

**Nioma0nwat3r**#161

YAY! Last one before Iām just going to link you to a Squeeze Theorem video because I hate explaining it.

Last one is Common Denominator. Use this one when you have a fraction for a numerator.

Hereās our equation:

What we need to do is get rid of the double fraction.

So letās multiply the numerator by the common denominator! Again, use a fractional one, and do a separate equation for each expression on the top. Remember to distribute the subtraction sign if you were originally subtracting the equations.

Because of fraction rules, you can multiply the numerator by (1/denominator), or in this case (1/x). From there, you can cancel the 'xās and get the new equation. Use direct substitution, and you get: -1/49