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4_4 - x may or may not exist and is called an indeterminate...

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Section 4.4: Indeterminate Forms and L’Hospital’s Rule Instructor: Ms. Hoa Nguyen ([email protected]) Indeterminate Forms 1) If lim x a f ( x ) = 0 and lim x a g ( x ) = 0, then lim x a f ( x ) g ( x ) may or may not exist, and is called an indeterminate form of type 0 0 . 2) If lim x a f ( x ) = ±∞ and lim x a g ( x ) = ±∞ , then lim x a f ( x ) g ( x ) may or may not exist, and is called an indeterminate form of type . Notice : f · g = f 1 g L’Hospital Rule Suppose f and g are differentiable and g 0 ( x ) 6 = 0 near a . If lim x a f ( x ) g ( x ) is an indeterminate form of type 0 0 or , then lim x a f ( x ) g ( x ) = lim x a f 0 ( x ) g 0 ( x ) if lim x a f 0 ( x ) g 0 ( x ) exists. Remarks : L’Hospital Rule is valid for “ x a ± ” or “ x → ±∞ ”. Example 1 of Section 4.4 (textbook): Example 2 of Section 4.4 (textbook): Example 3 of Section 4.4 (textbook): Example 4 of Section 4.4 (textbook): Example 5 of Section 4.4 (textbook): Example 6 of Section 4.4 (textbook): Example 7 of Section 4.4 (textbook): Indeterminate Powers 1) If lim x a f ( x ) = 0 and lim x a g ( x ) = 0, then lim x a f ( x ) g
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Unformatted text preview: ( x ) may or may not exist, and is called an indeterminate form of type . 2) If lim x → a f ( x ) = ∞ and lim x → a g ( x ) = 0, then lim x → a f ( x ) g ( x ) may or may not exist, and is called an indeterminate form of type ∞ . 3) If lim x → a f ( x ) = 1 and lim x → a g ( x ) = ±∞ , then lim x → a f ( x ) g ( x ) may or may not exist, and is called an indeterminate form of type 1 ±∞ . To calculate lim x → a f ( x ) g ( x ) of indeterminate-power forms, we can: • take the natural logarithm: y = f ( x ) g ( x ) ⇒ ln y = g ( x ) ln f ( x ), • or, write the function as an exponential: f ( x ) g ( x ) = e g ( x ) ln f ( x ) . then use L’Hospital’s Rule if possible. Example 8 of Section 4.4 (textbook): Example 9 of Section 4.4 (textbook): 1...
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