Definition of Limit Lesson 2-2

Verbal Definition of Limit L is the limit of f (x) as x approaches c if and only if L is the one number you can keep f (x) arbitrarily close to just by keeping x close to c, but not equal to c.

Example: A rational algebraic function Graph it! Yes! The denominator would go to zero. Then you would be dividing by zero. Yuck!!!! Hint Trace to x = -2 Why is there no value for the function at x = -2?

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?

Explore f (x) at x = -2 Tabular y 1 -2.03 1.94 -2.02 1.96 -2.01 1.98 -2. undef -1.99 2.02 In TBLSET set tblStart….-2.04 Δ……..0.01 Press ENTER. Then view the TABLE. Tabular Looking at the table, what number does f (x) appear to be at x = -2? Approaches 2

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. x y 1 -2.03 1.94 -2.02 1.96 -2.01 1.98 -2. undef -1.99 2.02 How about finding the limit algebraically?

Finding a limit algebraically… Remember For the original equation, x at -2 does not exist. We say its approaching a value of 2.

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.

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: -1 The 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.

Formal Definition of Limit: L is the limit of f (x) as x approaches c if and only if for any positive number epsilon (ε), no matter how small, there is a positive number delta (δ) such that if x is within delta units of c (but not equal to c) then f (x) is within epsilon units of L.

Formal Definition of Limit L is the limit of f (x) as x approaches c if and only if for any positive number ε, no matter how small, L is 4 c is 2 ε is 1 “arbitrarily” δ is 0.8 and 0.6 there is a positive number δ such that if x is within δ units of c (but not equal to c) then f (x) is within ε units of L Pick the δ that is most restrictive. So δ is 0.6

Infinite Discontinuity What is the limit at x = 2? Infinite Discontinuity Answer: No limit Removable Discontinuity What 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.

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: 7 0/0 called Indeterminate Form Try to evaluate f (1) by direct substitution. What form does the answer take? What is this form called?

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 7 as x approaches 1 Ω