Lesson: Limits of functions

Direction: Pick-up lines regarding limit of function At this point, make an output by making a DIY bookmark with pick-up lines/puns regarding the limit of a function. You can use recyclable decorative materials available at home. The standard size of your bookmark must be 2-inch by 7-inch. The scoring rubric on the next page will be used in assessing your outputs. Example: "If my love for you were an equation, it would be lim ??2 1 ??2 because it has no limit.

Additional Information that will help you in making pick-up lines/puns, click the pictures below.

Discover Today, you will follow the travels of Benny and Bertha Bug. With their help, we will look at graphs of rational functions and piecewise functions from a bug's eye view to help convey the important concept of limits in an intuitive way. 31:1\"sz / wwwpinterestph/pin/ 646336983999465697/ The concept of a \"limit\" is the building block on which all the underlying concepts of calculus are based. It helps us to describe, in precise way, the behavior of Jx) when x is close, but not equal, to a particular value 6. Limits are the backbone of calculus, and calculus is called the Mathematics of Change. To visualize it further, imagine that you are going to watch a basketball game. When you choose seats, you would want to be as close to the action as possible. You would want to be as close to the players as possible and have the best view of the game, as if you were in the basketball court yourself. Take note that you cannot actually be in the court and join the players, but you will be close enough to describe clearly what is happening in the game. This is how it is with limits of functions. We will consider functions of a single variable and study the behavior of the function as its variable approaches a particular value (a constant}. The variable can only take values very, very close to the constant, but it cannot equal the constant itself. However the limit will be able to describe clearly what is happening to the function near the constant. Before going to the formal denition of alimit of a function let us observe the behavior of the given function as x approaches a value. Denition oi the Limit oi a Function Let f be a function dened at every number in some open interval containing c, except possibly at the number citself. If the value of f is arbitrarily close to the number L for all the values of at sufciently close to c, then the limit of ffx) as x approaches c is I... This is written as tho f(x) = L. We also have a special notation to talk about limits. For instance, this is how we would write the limit of f as x approaches 3: \"The limit of...\" \"...the function f...\" lim f(x) / \"...as x approaches 3.\"'-""'-"J.r \"it)3 The symbol lim means we are taking a limit of something. The expression to the right of the lim is the expression we are taking the limit of. In our case, that's the function f. The expression x - 3 that comes below the lim means that we take the limit of f values of x approach 3. EXAMPLE 1: What is the lim (x + 2)? Solution: The arrow pointing at 4 indicates that x is approaching 4 from the left side and from the right side. This means that x can take on values less than 4 and values greater than 4. It cannot take as a value because it is just approaching 4. The first thing to do to find the limit of the given function is to construct the table of values. x approaches4 from the left x approaches4 from the right X 3.9 3.99 3.999 3.9999 4 4.0001 4.001 4.01 4.1 fix ) 5.9 5.99 5.999 5.9999 6.0001 6.001 6.01 6. 1 f( x) approaches6 f( x) approaches6 As the value of x gets closer to 4 from the left or as x approaches 4 from the right, the value of f(x) approaches 6. In other words, the value of f(x) gets closer and closer to 6 as the value of x gets closer and closer to 4 either side. This can be written as : lim(x + 2) = 6 This means that the limit of (x + 2) is 6 as x approaches 4 from either side. EXAMPLE 2: Evaluate the limit of lim (x2 + 1). x--1 Here c = -1 and f(x) = x2 + 1. We construct table of values approaching -1 from the left and approaching -1 from the right. x approaches -1 from the left x approaches -1 from the right X -1.2 -1.01 -1.001 -1 0.9999 -0.99 -0.8 f (x) 2.44 2.0201 2.00020001 2 1.99980001 1.9801 1.64 f(x) approaches 2 f(x) approaches 2 The table show that as x approaches -1, f(x) approaches 2. In symbols,lim (x2 + 1) = 2. 161 Have you noticed a pattern in the way we have been investigating a limit? We have been specifying whether x will approach a value c from the left, through values less than c, or from the right, through the values greater that c. This direction may be specied in the limit notation, lim f(x) by adding certain symbols. x)C . If x approaches 6 from the left, or through values less than c, then we write 11m, f(x). x)C . If x approaches 6 from the right, or through values greater than c, then we write lim+ f(x). x)C Furthermore, we say lim f(x) = L x)C if and only if 1im_ for) = L and lim+ f(x) = L xH: x)(,' In other words, for the limit L to exist, the limits from the left and from the right must both exist and be equal to L. Therefore, lim f(x) DNE' whenever 1im_ x) at 1im+ x). 1411' These limits, limi Jx) and lim+ x), are also referred to as one-sided limits, x)C x)C since you only consider values on one side of 6. EXAMPLE 3: Ehraluate the limit of 133 x) given its graph. 3 f I I I I The given graph is a graph of a piecewise function. The limit of the given function as x approaches 0 is 2 because if we're going to take a look at the values of x from the left: and the values of x from the right, both directions approaches 2 even ifit is evident that f(0) = 1. Still the value being approached by both directions is 2. Thus, we can say that Eggx) = 2 Limit and Function Value The limit of a function as it approaches c is not necessarily equal to its e value. Thus. lim f (x) can assume a value different from c). xIf Solution: In the given function f, the limit does not exist because f(0) is undened and as x moves closer to 0, the function approaches two different values. The specied limit does not exist (DNE). In symbols, 53% at) DNE 3.931310. IDEA We need to emphasize an important fact. We do not say that \"1310 f(x) I \"equals DNE\Solution: The process is still the same even though it looks a bit different since the given function is a piecewise function. We still approach the constant 4 from the left and right, but take note that we should evaluate the appropriate corresponding functional expression. In this case when x approaches 4 from the left, the values taken should be substituted in f(x) = x + 1. Indeed, this is the part of the function which accepts values less than 4. On the other hand, when x approaches 4 from the right, the values taken should be substituted in f (x) = (x -4) 2 + 3. So, x approaches 4 from the left x approaches 4 from the right X 3.9 3.99 3.999 3.9999 4 4.0001 4.001 4.01 4.1 fix ) 4.9 4.99 4.999 4.9999 -3 3.0001 3.001 3.01 3.1 fix) approaches 5 f(x) approaches 3 Observe that the values that f(x) approaches are not equal, namely f(x) approaches 5 from the left while it approaches 3 from the right. In such a case, we say that the limit of the given function does not exist (DNE). In symbols, lim f (x) DNE. EXAMPLE 6: Evaluate lim -4, x-2 x-2 numerically and graphically. Solution: Set up a table of values from both sides of x = 2 and find the value that the function approaches from both directions. x approaches 2 from the left x approaches 2 from the right x 1.9 1.99 1.999 2 2.001 2.01 2.1 f(x) 3.9 3.99 3.999 undefined 4.001 4.01 4.1 f(x) approaches 4 f(x) approaches 4 Thus, lim x 2 - 4. =4. This is shown in the graph below.Observe that it doesn't matter if f (2) is undefined. The function can still have a limit, as long as it approaches the same real number from the left and from the right. BIG IDEA The limit of a function at a specified value of x gives us a value to which it is not possible to go beyond. Similarly, we have our own limitations. We are restricted to do things beyond our human capacities