lecture14hout

# lecture14hout - MATH 006 Calculus and Linear...

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MATH 006 Calculus and Linear Algebra (Lecture 14) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 17

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Outline 1 Algebraic Methods for Finding Limits 2 Indeterminate Form 3 Limits of the Difference Quotient Albert Ku (HKUST) MATH 006 2 / 17
Algebraic Methods for Finding Limits Algebraic Methods for Finding Limits From the properties of limits, we have the following fact: Suppose f ( x ) is a function which is defined by an expression that involves addition, subtraction, multiplication, division and/or powers of x . If f ( c ) is well-defined, then lim x c f ( x ) = lim x c - f ( x ) = lim x c + f ( x ) = f ( c ) Remark If f ( c ) is not well-defined, it does not necessarily mean that lim x c f ( x ) does not exist. We will see more examples later. Albert Ku (HKUST) MATH 006 3 / 17

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Algebraic Methods for Finding Limits Example Let f ( x ) = 3 x + 1 - 2 x 2 5 3 x 2 - 2 x - 1 . Find lim x 2 f ( x ). Solution Since f (2) = 3 3 - 8 5 7 is well-defined, lim x 2 f ( x ) = f (2) = 3 3 - 8 5 7 Albert Ku (HKUST) MATH 006 4 / 17
Algebraic Methods for Finding Limits Example Let f ( x ) = 3 x + 1 - 2 x 2 5 3 x 2 - 2 x - 1 . Find lim x 1 f ( x ). Notice that as x tends to 1, the denominator will get close to 0 but the numerator will get close to 3 2 - 2 6 = 0. You would expect the value of f will get larger and larger (positively or negatively, depending on the direction of approach) and will not attain any finite limit. In fact, we have

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