Limits in Calculus

Unit 1: Limits THE FOUNDATION OF CALCULUS

Lessons 1.2 Definition of a Limit (Graphically and Numerically) 1.3A Evaluating Limits Analytically 1.3B Properties of Limits 1.3C Special Trig Limits 1.5 Infinite Limits (Vertical Asymptotes) 3.5 Limits at Infinity (End Behavior) Review on Limits 1.4 Continuity 1.5 Continuity and Limits Review

1.2 Definition of a Limit WHAT IS A LIMIT?

The Idea of a Limit The graph of f(x)=2x+3 is shown to the right. What happens to f(x) as x gets close to 3? From the left From the Right

The Idea of a Limit Since in our previous example, as x got closer and closer to 3 from the left and right hand sides, the y value got closer and closer to 9, we say that ( ) 9 3 = + lim x 2 x The limit of f(x)=2x+3 as x approaches 3 is equal to 9 or 3

Definition of a Limit As x approaches a, the limit of f(x) is L = lim ( ) f x L x a As x gets close to some number a, y (or f(x)) is approaching some number L. Limits give us an idea of what y-values graphs are heading towards around certain x values.

One Sided Limits = lim ( ) f x L The limit as x approaches a from the left x a = lim ( ) f x L The limit as x approaches a from the right + x a For a limit to exist both the limit from the left AND the right must be the same

2 4 x = lim x Find 2 2x Step 1: Find the y-value as x approaches 2 from the left 2 4 x 2 4 x = ( ) f x = lim x 2 x 2 2 x Step 2: Find the y-value as x approaches 2 from the right 2 4 x = lim x 2 2 x + Step 3: If both y-values are the same, that y-value is the limit!

Find Each Limit = = lim x ( ) f x = lim x ( ) f x = lim x ( ) g x lim x ( ) g x 1 0 2 0

Examples = 1 = lim x ( ) f x lim x ( ) f x 1) 5) 4 = = ) 1 ( f ) 4 ( f 2) 6) = = lim x ( ) f x lim x ( ) f x 3) 7) + 2 2 = = lim x ( ) f x lim x ( ) f x 4) 8) 2 3

When a Limit Does Not Exist 1) If a limit approaches different y-values from the left and right then the limit a x lim ( ) lim x ( ) f x f x does not exist. + a 2) Limits that go to infinity also do not exist. Write the answer as or - .

In summary lim x ( ) f x 1) The limit of f(x) as x approaches some number a is written as a 2) The answer to a limit problem is a) A y-value: If the graph approaches the same y-value from the left and right b) DNE: If the graph does not approach the same y-value from the left and right c) if both sides head towards negative or positive infinity from the left and right. Limits that go towards infinity Do Not Exist.

Homework HW 1.2: pg 54-58 #9-16, 25, 26, 49-52 (Just graph to find limits, don't write paragraph)

1.3A Evaluating Limits Analytically HOW DO WE EVALUATE LIMITS USING ALGEBRA?

Another way to find limits We don t always have a nice graph of the function readily available to us. When this is the case oftentimes it is helpful to take limits without having to graph them. 2 x lim x 2 3

Methods we will talk about Today Plan A: Direct Substitution Plan B: Algebraic Simplification Plan C: Multiply by Conjugate (Rationalizing) Plan D: Piecewise Functions

Plan A: Direct Substitution Just plug in the value for the limit and see if you get a defined value! ( )= 1 2 x = + = 3 lim x 2 x lim x 2 x x lim x 2 11 1 3 2 25 x = lim x + 5 5 x

Plan B: Algebraic Simplification 0 If you end up with try to factor and get something to cancel so you can use direct substitution. 0 x + 2 3 2 x x 2 25 x = lim x = lim x 1 + 1 5 5 x x 2 3 2 5 x x = * lim x 1 2

A hole or removable discontinuity Removable Discontinuity: A point at which a graph is not connected but can be made connected by filling in a single point. (Same as a hole on a graph) 2 25 x = ( ) f x + 5 x How can we rewrite f(x) as a simpler function? f(x)=___________

Plan C: Multiply by Conjugate (Rationalizing the Numerator) 0 If you end up with and there is nothing to factor, try multiplying by the conjugate if you have square roots in the numerator. 0 + 1 3 x 4 3 x = = lim x lim x 8 x 13 x 8 13

Plan D: Piecewise Functions When taking limits of a piecewise function, use direct substitution on both parts of the graph if you want to take the limit of where the function switches over. ( lim 6 x = = lim x ( ) f x ) f x + 2 , 6 x x 8 = ( ) f x 2 1 , 6 x x = lim x ( ) f x + 6 = lim x ( ) f x 6

Practice: Evaluate Each Limit + 2 3 , 6 , + 3 3 x x x = g x ( ) g x 2 x + = lim x 8 3 x 1) = lim ( ) x 9 3) 3 + 2 2 5 3 3 x x x = = lim x lim x 2) 2 2 3 9 x x x 4) 3 2 9

In Summary Plan A: Direct Substitution: Plug in the x value (may not work) Plan B: Algebraic Simplification: Factor and Simplify etc. so that x- values can be plugged in Plan C: Multiply by Conjugate (Rationalizing the Numerator): Multiply by conjugate of numerator and cancel so that x-values can be plugged in. (Use if you see square roots) Plan D: Piecewise Functions: Make sure to use direct substitution from left AND right hand sides if taking the limit of a place where the function switches over.

Homework HW 1.3A: pg 67-68 #1, 4, 9, 13, 15, 17, 23, 41- 43, 49, 50, 51, 52, 55

1.3B Properties of Limits WHAT ARE SOME PROPERTIES OF LIMITS? HOW DO WE WORK WITH X LIMITS APPROACHING 0?

= = Properties of Limits lim ( ) f x L lim ( ) g x M x a x a 1) Sum Rule: The limit of a sum of two functions equals the sum of their limits x g x f a x + = + lim ( ) ( ) L M 2) Difference Rule: The limit of a Difference of two functions equals the difference of their limits x g x f a x ( x x 3 Vs. = lim ( ) ( ) L M ( ) x ) x ( ) x 2 2 lim lim lim 3 3 x x

= = Properties of Limits lim ( ) f x L lim ( ) g x M x a x a 3) Product: The limit of a product of two functions equals the product of their limits x g x f a x = lim ( ) ( ) L M 4) Quotient: The limit of a quotient of two functions equals the quotient of their limits L x g ) ( ( ) f x = lim M x a 5) Constant Rule: The limit of a constant times a function is the constant times the limit of the function. k x f k a x = lim ( ) L

Examples = lim x ( ) 5 g x = Lets say that and lim x ( ) 2 f x a a = lim x ( ) ( ) f x g x 3) a Find ( ) f x = + lim x ( ) ( ) f x g x = lim x 1) 4) ( ) g x a a = = lim x ( ) ( ) f x g x lim x 6 ( ) g x 2) 5) a a

+ ( ) ( ) f x x f x lim x 0 x Delta X Notation For Limits x f ( lim 0 + ) ( ) x f x If f(x) = 4x-1 find: Idea: First find f(x+ x). Write it down. x x Plug in f(x+ x) and f(x) into the formula. Be careful with parenthesis. Since plugging in 0 for x gives us a 0 in the denominator we need to do some algebra. The x should cancel. Note: x is a different variable than x.

Remember that x and x are different variables. You may end up with x sin your answer but not x s since you should be Substituting 0 in for x. Examples + ( ) ( ) f x x f x lim Find for each function. x 0 x 2) f(x) = x2 3) f(x) = x3 1) f(x)= -2x+6

= = lim ( ) g x M lim ( ) f x L x a x a In Summary + = + lim ( ) ( ) f x g x L M 1) Sum: Simplifying Limits in the Delta X Formula x f + ( lim 0 x a ) ( ) x f x = lim ( ) ( ) f x g x L M 2) Difference: x x x a 1) To get the numerator, plug in x-x into the function to find f( x-x) Then subtract the original function. = lim ( ) ( ) f x g x L M 3) Product: x a ( ) f x L = lim 4) Quotient: ( ) g x M x a 2) Since plugging in 0 for x gives us a 0 in the denominator we need to do some algebra to find the limit. = lim ( ) k f x k L 5) Constant: x a

Homework HW 1.3B: pg 67-69 #18,25,37,45-48, 56, 59, 83, 85, 113, 114, 116, 117

1.3C Special Trig Limits WHAT ARE SOME SPECIAL TRIG LIMITS WE NEED TO KNOW? WHAT TRIG IS MOST IMPORTANT TO REMEMBER FOR THE AP TEST?

Know your Unit Circle There will be questions that will require unit circle knowledge throughout calculus. If you don t know them you will need to study! = lim tan x = = lim cot x 5) lim sin x 3) 1) 7 3 x x x 6 4 2 = lim csc x = lim cos x 6) = lim sec x 4) 2) 3 x x x 6 4

Know your Unit Circle ANSWERS ANSWERS There will be questions that will require unit circle knowledge throughout calculus. If you don t know them you will need to study! 1 3 = lim x tan x or = = 5) lim x cot 1 x lim x sin 1 x 3) 1) 3 3 7 3 6 4 2 2 = lim x csc 2 = x lim x sec 1 x 6) = 4) lim x cos x 2) 3 2 6 4

Most Important Trig Identities To Remember sin x = tan x + = 2 2 sin cos 1 x x cos x = sin 2 2 sin cos x x x

Special Trig Limits Need to memorize these for AP Test! Will be used in many trig limit problems sin x 1 cos x x = lim x 1 = lim x 0 x 0 0

Examples x sin 3 lim 0 sin 2 x = 1) 4) = lim x x 5 x 0 x + 2 2 sin cos cos x x x 1 sin cot = = lim lim 2) 5) x 0 x 0 sin 3 x x = lim = lim 3) 6) x 0 x sin x 0 x

Examples ANSWERS ANSWERS sin 3 lim 0 x x sin 2 x 2 x = 3 = 1) 4) lim x x 5 5 0 + 2 2 sin cos cos x x x 1 sin cot = = lim x 0 lim 0 2) 5) x 0 0 sin 3 x x = lim x 3 = lim x 1 3) 6) x 0 sin x 0

In Summary Make sure you are confident with your trig. If you need to make flash cards to review, do so! Don t forget these 2 special limits. Be careful! They only apply if x approaches 0. 1 cos x x sin x sin x = = lim x 0 lim x 1 = lim x 0 0 x x

Homework HW 1.3C: pg 67-69 #3, 27-36, 67-75, 77

1.5 Infinite Limits WHAT ARE SOME PROPERTIES OF LIMITS? HOW DO WE WORK WITH X LIMITS APPROACHING 0?

Find the Vertical Asymptotes for Each Function =7 1 x x 3 x 1) = 4) ( ) f x ( ) f x + 2 5 6 x x 1 = ( ) f x 2) 2+ 5 x x + 3 x =x ( ) f x 3) 2 2

Find the Vertical Asymptotes for Each Function ANSWERS ANSWERS =7 x=7 1 x x 3 x 1) = 4) ( ) f x ( ) f x + 2 5 6 x x x= -6 1 = x=0, x= -5 ( ) f x 2) 2+ 5 x x + 3 x =x ( ) f x 3) x= 2 2 2

Finding Limits at Infinity using a Graphing Calculator lim 2 x 1 1 = = lim x 2 2 2 x + 1) Using a Graph Input the function in y= Observe the graph x 1 = lim x 2 2x 2) Using the table feature-> Go to Setup Table start: Choose the value of the asymptote Set tbl to a small value like .01 and save. Then view the table. Observe the values on the left and right side of the asymptote

Graphing Calculator Examples 3 1 = = lim x lim ( ) 2 + 4 x 2 x 2 4 x