11.1 Introduction to Limits Module 11: Limits 11.1 Introduction to Limits

What is a Limit? In order to understand calculus, you need to know what a “limit” is. A limit is the value a function approaches as the variable within that function gets nearer and nearer to a particular value. In other words. What y value does our function approach as x approaches a different value.

Limit example What is the limit of the function 𝑓 𝑥 = 𝑥 2 as x approaches 2? In limit notation we write it like this. lim 𝑥→2 𝑓(𝑥) . In order to figure this out we need to test values that are closer and closer to 2. When x =1.9, f(x) = 3.61 When x =1.99, f(x)=3.9601 When x=1.999, f(x)=3.999 As x increases and approaches 2, f(x) gets closer and closer to 4. This is called the left handed limit and is written lim 𝑥→ 2 − 𝑓(𝑥)

Limit Example Continued What is the limit when the x is bigger than 2? When x = 2.1, f(x) = 4.41 When x = 2.01, f(x) = 4.0401 When x = 2.0001, f(x) = 4.00040001 As x decreases and approaches 2, f(x) still approaches 4. This is called the right-hand limit and is written like this: lim 𝑥→ 2 + 𝑓(𝑥)

We got the same answer when evaluating both the left- and right- hand limits, because when x is 2, f(x) is 4. You should always check both sides of the limit because sometimes you don’t get the same answer. We write our final answer as follows: lim 𝑥→2 𝑥 2 =4

Theorem 2 Polynomial and Rational Functions If f(x) is any polynomial function and c is any real number, then lim 𝑥→𝑐 𝑓 𝑥 =𝑓(𝑐) If f(x) and g(x) are polynomials and c is any real number, then lim 𝑥→𝑐 𝑓 𝑥 𝑔 𝑥 = 𝑓 𝑐 𝑔 𝑐 , provided that 𝑔(𝑐)≠0

Limit Example Find lim 𝑥→5 𝑥 2 The approach is simple: Plug in 5 for x, and you get 25.

Limit Example using theorem 2

Examples of Limit Properties Example: lim 𝑥→5 3 𝑥 2 3 lim 𝑥→5 𝑥 2 =75

Examples of Limit Properties lim 𝑥→5 𝑥 2 + lim 𝑥→5 𝑥 3 =150

Examples of Limit Properties lim 𝑥→5 𝑥 2 +1 lim 𝑥→5 𝑥−1 =52