tailieunhanh - Chapter 5: Inverse functinons – Section 5.8: Indeterminate forms and l’hospital’s rule

Mời các bạn tham khảo bài giảng Chapter 5: Inverse functinons – Section : Indeterminate forms and l’hospital’s rule sau đây. Bài giảng dành cho đối tượng sinh viên ngành Công nghệ thông tin. Tham khảo nội dung bài giảng để nắm bắt nội dung chi tiết. | SECTION INDETERMINATE FORMS AND L’HOSPITAL’S RULE P INDETERMINATE FORMS Suppose we are trying to analyze the behavior of the function Although F is not defined when x = 1, we need to know how F behaves near 1. In particular, we would like to know the value of the limit P INDETERMINATE FORMS In computing this limit, we can’t apply Law 5 of limits (Section ) because the limit of the denominator is 0. In fact, although the limit in Expression 1 exists, its value is not obvious because both numerator and denominator approach 0 and is not defined. P INDETERMINATE FORM —TYPE 0/0 In general, if we have a limit of the form where both f(x) → 0 and g(x) → 0 as x → a, then this limit may or may not exist. It is called an indeterminate form of type . We met some limits of this type in Chapter 1. P INDETERMINATE FORMS For rational functions, we can cancel common factors: We used a geometric argument to show that: P INDETERMINATE FORMS However, these . | SECTION INDETERMINATE FORMS AND L’HOSPITAL’S RULE P INDETERMINATE FORMS Suppose we are trying to analyze the behavior of the function Although F is not defined when x = 1, we need to know how F behaves near 1. In particular, we would like to know the value of the limit P INDETERMINATE FORMS In computing this limit, we can’t apply Law 5 of limits (Section ) because the limit of the denominator is 0. In fact, although the limit in Expression 1 exists, its value is not obvious because both numerator and denominator approach 0 and is not defined. P INDETERMINATE FORM —TYPE 0/0 In general, if we have a limit of the form where both f(x) → 0 and g(x) → 0 as x → a, then this limit may or may not exist. It is called an indeterminate form of type . We met some limits of this type in Chapter 1. P INDETERMINATE FORMS For rational functions, we can cancel common factors: We used a geometric argument to show that: P INDETERMINATE FORMS However, these methods do not work for limits such as Expression 1. Hence, in this section, we introduce a systematic method, known as l’Hospital’s Rule, for the evaluation of indeterminate forms. Another situation in which a limit is not obvious occurs when we look for a horizontal asymptote of F and need to evaluate the limit P INDETERMINATE FORMS It isn’t obvious how to evaluate this limit because both numerator and denominator become large as x → ∞. There is a struggle between the two. If the numerator wins, the limit will be ∞. If the denominator wins, the answer will be 0. Alternatively, there may be some compromise—the answer may be some finite positive number. P INDETERMINATE FORM —TYPE ∞/∞ In general, if we have a limit of the form where both f(x) → ∞ (or – ∞) and g(x) → ∞ (or – ∞), then the limit may or may not exist. It is called an indeterminate form of type ∞/∞. P INDETERMINATE FORMS We saw in Section that this type of limit can be evaluated for certain .