lect116_4_f11

lect116_4_f11 - Friday September 16 Lecture 4 Limits(Refers...

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Friday September 16 Lecture 4 : Limits (Refers to Section 2.1 to 2.3 in your text) Students who have mastered the content of this lecture know about : An intuitive definition of limits of functions , one-sided limits , infinite limits , limits at infinity , vertical asymptotes , horizontal asymptotes , limit laws , the squeeze theorem , the limit of sin(x)/x as x tends to 0 is 1 . Students who have practiced the techniques presented in this lecture will be able to : Use limit laws and use elementary techniques to compute values of limits. 4.1 Definition ( Intuitive definition ) Suppose we are given a function f ( x ) and a number a . When we say that the “ limit of f ( x ) as x approaches a is L ” written as lim x a f ( x ) = L we mean that “ f ( x ) becomes arbitrarily close to L as x approaches a from either side”. (Note: There is a formal definition of the limit of a function in section 2.4 of your text. This is optional reading. We will not present this in class nor is it required for Math 116 students to know this.) 4.1.1 Example – For the function f ( x ) whose curve is illustrated below see that the limit of f ( x ) as x approaches 1 from either side is 6, while the limit of f ( x ) as x approaches 1 from either side is 3. 4.1.2 Examples : For each of the following we can determine the limits simply my visualizing the curve of these functions. a) lim x 2 x 2 = 4 b) lim x a x = a. c) lim x 0 (1/ x ) does not exist. (That is, the function 1/ x does not approach a number when x approaches 0.) d) If c is a number and f ( x ) = c for all x , then lim x a f ( x ) = c for any a . Equivalently. lim x a c = c
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4.2 Definition – One-sided limits . Say we are given a function f ( x ) and a number a . We define the left-handed limit as follows: It is the limit of the function when the value of x approaches a only from the left side. It is written as lim x a f ( x ) = L We define the right-handed limit as follows: It is the limit of the function when the value of x approaches a only from the right side. The expression “limit of f ( x ) as x approaches a from the right is L ” is written as lim x a + f ( x ) = L Plotting the curve of a function often helps us determine left-handed and right-handed limits. Note : Sometimes there is no number L which is the limit of a function when x approaches a number a . When this is the case we say that the limit does not exist . In the following graph of a curve
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lect116_4_f11 - Friday September 16 Lecture 4 Limits(Refers...

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