In your own words: What is a limit?

Evaluating Limits Analytically Evaluate a limit using properties of limits Develop and use a strategy or finding limits Evaluate a limit using dividing out and rationalizing techniques Evaluate a limit using the Squeeze Theorem

Some Basic Limits Let 𝑏 and 𝑐 be real numbers and let 𝑛 be a positive integer. lim 𝑥→𝑐 𝑏 =𝑏 lim 𝑥→𝑐 𝑥 =𝑐 lim 𝑥→𝑐 𝑥 𝑛 = 𝑐 𝑛

Properties of Limits Let 𝑏 and 𝑐 be real numbers, let 𝑛 be a positive integer, and let 𝑓 and 𝑔 be functions with the following limits lim 𝑥→𝑐 𝑓 𝑥 =𝐿 and lim 𝑥→𝑐 𝑔 𝑥 =𝐾. Scalar Multiple lim 𝑥→𝑐 𝑏𝑓 𝑥 =𝑏𝐿 Sum or Difference lim 𝑥→𝑐 𝑓 𝑥 ±𝑔 𝑥 =𝐿±𝐾 Product lim 𝑥→𝑐 𝑓 𝑥 𝑔 𝑥 =𝐿𝐾 Quotient lim 𝑥→𝑐 𝑓 𝑥 𝑔 𝑥 = 𝐿 𝐾 𝐾≠0 Power lim 𝑥→𝑐 𝑓 𝑥 𝑛 = 𝐿 𝑛

lim 𝑥→3 (2 𝑥 3 +3𝑥−4)

Limits of Polynomial and Rational functions If 𝑝 is a polynomial function and 𝑐 is a real number, then lim 𝑥→𝑐 𝑝 𝑥 =𝑝 𝑐 . If 𝑟 is a rational function given by 𝑟 𝑥 = 𝑝 𝑥 𝑞 𝑥 and 𝑐 is a real number such that 𝑞(𝑐)≠0, then lim 𝑥→𝑐 𝑟 𝑥 =𝑟 𝑐 = 𝑝 𝑐 𝑞 𝑐 .

The Limit of a Function Involving a Radical Let 𝑛 be a positive integer. The following limit is valid for all 𝑐 if 𝑛 is odd, and is valid for 𝑐>0 if 𝑛 is even. lim 𝑥→𝑐 𝑛 𝑥 = 𝑛 𝑐

The Limit of a Composite Function If 𝑓 and 𝑔 are functions such that lim 𝑥→𝑐 𝑔 𝑥 =𝐿 and lim 𝑥→𝐿 𝑓 𝑥 =𝑓 𝐿 , then lim 𝑥→𝑐 𝑓 𝑔 𝑥 =𝑓 lim 𝑥→𝑐 𝑔 𝑥 =𝑓 𝐿 .

So… Hopefully you’ve figured out by now, to find the limit, just put the 𝑐 value into the equation… That’s it. It works for all trig functions too. So, like, lim 𝑥→𝑐 sin 𝑥 = sin 𝑐 . I’m not going to make you copy all of them down, unless you want to.

Functions That Agree at All But One Point Let 𝑐 be a real number and let 𝑓 𝑥 =𝑔 𝑥 for all 𝑥≠𝑐 in an open interval containing 𝑐. If the limit of 𝑔 𝑥 as 𝑥 approaches 𝑐 exists, then the limit of 𝑓 𝑥 also exists and lim 𝑥→𝑐 𝑓 𝑥 = lim 𝑥→𝑐 𝑔 𝑥 .

Functions That Agree at All But One Point - Simplified This is a fancy way of saying that you can manipulate a function, without changing it, to find a value that, otherwise, wouldn’t exist.

lim 𝑥→4 𝑥 2 −𝑥−12 𝑥−4

lim 𝑥→0 𝑥−2 +2 𝑥

The Squeeze Theorem If ℎ 𝑥 ≤𝑓 𝑥 ≤𝑔 𝑥 for all 𝑥 in an open interval containing 𝑐, except possibly at 𝑐 itself, and if lim 𝑥→𝑐 ℎ 𝑥 = lim 𝑥→𝑐 𝑔 𝑥 =𝐿 then lim 𝑥→𝑐 𝑓 𝑥 =𝐿.

lim 𝑥→0 sin 𝑥 𝑥

lim 𝑥→0 1− cos 𝑥 𝑥