2 Verbal Definition of Limit L is the limit of f (x) as x approaches cif and only ifL is the one number you can keep f (x)arbitrarily close to just by keeping x close to c, but not equal to c.
3 Example: A rational algebraic function Graph it!Yes! The denominator would go to zero. Then you would be dividing by zero. Yuck!!!!HintTrace to x = -2Why is there no value for the function at x = -2?
4 Explore f (x) at x = -2 Algebraically, Indeterminate Form Looking back at the graph, what number does f (x) appear to be at x = -2?
5 Explore f (x) at x = -2 Tabular y 1-2.031.94-2.021.96-2.011.98-2.undef-1.992.02In TBLSET set tblStart… Δ……..0.01Press ENTER. Then view the TABLE.TabularLooking at the table, what number does f (x) appear to be at x = -2?Approaches 2
6 How about finding the limit algebraically? Looking at both graph and table of f(x) we can conclude the limit as x approaches -2 is 2.xy 1-2.031.94-2.021.96-2.011.98-2.undef-1.992.02How about finding the limit algebraically?
7 Finding a limit algebraically… RememberFor the original equation,x at -2 does not exist. We say its approaching a value of 2.
8 Removable Discontinuity The function is discontinuous because of the gap at x = -2 but the gap can be removed by defining f (-2) to be 2.An open circle at the point of discontinuity is used to illustrate that.
9 Step Discontinuity This function is discontinuous because of the gap at x = 0.However, it can not simply be removed since there is a large “step” between the two branches.What is the limit at x = 2?What is the limit at x = 0?Answer: -1The discontinuity is at x = 0. As x approaches 2 from the left, g(x) is close to -1. As x approaches 2 from the right, g(x) is still close to -1.Answer:Does not exist.
10 Formal Definition of Limit: L is the limit of f (x) as x approaches cif and only iffor any positive number epsilon (ε),no matter how small, there is a positivenumber delta (δ)such thatif x is within delta units of c (but not equal to c)then f (x) is within epsilon units of L.
11 Formal Definition of Limit L is the limit of f (x) as x approaches cif and only iffor any positive number ε,no matter how small,L is 4c is 2ε is 1 “arbitrarily”δ is 0.8 and 0.6there is a positive number δ such thatif x is within δ units of c (but not equal to c)then f (x) is within ε units of LPick the δ that is most restrictive.So δ is 0.6
12 Infinite Discontinuity What is the limit at x = 2?Infinite DiscontinuityAnswer: No limitRemovableDiscontinuityWhat is the limit at x = 5?Answer: 2 The value of h(5) is 4 but h(x) approaches 2 from the left and from the right.
13 Example: Given Answer: 7 0/0 called Indeterminate Form From the graph, what do you think the limit of f (x) is as x approaches 1?Answer: 70/0 calledIndeterminate FormTry to evaluate f (1) by direct substitution. What form does the answer take? What is this form called?
14 Limit is 7 as x approaches 1 Factor the numerator and simplify the expression. Although the simplified expression does not equal f (1), you can substitute 1 for x and get an answer. What is the answer and what does it represent?Limit is 7as x approaches 1Ω