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**Unformatted text preview: **MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Limits Dr. WONG Chi Wing Department of Mathematics, HKU MATH0201 BASIC CALCULUS Limits at a Point Two–sided and One–sided Limits Evaluation of Limits Indeterminate Form Continuity Continuous Functions Intermediate Value Theorems Infinite Limits and Limits at Infinity Infinite Limits Limits at Infinity Reference § 1.5–6 of the textbook. MATH0201 BASIC CALCULUS Limits at a Point Two–sided and One–sided Limits Definition 1 (Limit at a Point (p.64)) Let f be a function and c ∈ R . We write lim x → f ( x ) = L or f ( x ) → L as x → c and call L the limit of f at c if f ( x ) gets closer and closer to L as x gets closer and closer to c from both sides, but not equal to c . MATH0201 BASIC CALCULUS Limits at a Point Two–sided and One–sided Limits Example 2 (Graphical Approach) Consider the graph of f ( x ) = x , x 6 = 0. We have lim x → f ( x ) = 0. lim x →- x = lim x → + f ( x ) = . MATH0201 BASIC CALCULUS Limits at a Point Two–sided and One–sided Limits Example 3 (Graphical Approach) Consider the graph of f ( x ) = | x | / x , x 6 = 0. lim x → f ( x ) DOES NOT exist. lim x →- | x | x =- 1 and lim x → + | x | x = 1 . MATH0201 BASIC CALCULUS Limits at a Point Two–sided and One–sided Limits Definition 4 (One–sided Limit at a Point: Left Hand Limit (p.79)) Let f be a function and c ∈ R . We write lim x → c- f ( x ) = L and call L the left hand limit of f at c if f ( x ) gets closer and closer to L as x gets closer and closer to c from the left hand side, but not equal to c . MATH0201 BASIC CALCULUS Limits at a Point Two–sided and One–sided Limits Definition 5 (One–sided Limit at a Point: Right Hand Limit (p.79)) Let f be a function and c ∈ R . We write lim x → c + f ( x ) = L and call L the right hand limit of f at c if f ( x ) gets closer and closer to L as x gets closer and closer to c from the right hand side, but not equal to c . MATH0201 BASIC CALCULUS Limits at a Point Two–sided and One–sided Limits Remark 1 lim x → c f ( x ) may also be called an two–sided limits . Theorem 6 (Existence of a Limit (p.80)) The (two–sided) limit lim x → c f ( x ) exists and equals L if and only if both the one–sided limits lim x → c- f ( x ) and lim x → c + f ( x ) exist and they are equal to L. MATH0201 BASIC CALCULUS Limits at a Point Two–sided and One–sided Limits Example 7 Below is the graph a function g ( x ) . 1. lim x → 3 g ( x ) = 2. lim x → 1- g ( x ) = and lim x → 1 + g ( x ) = 3. lim x → 3- g ( x ) = 4. lim x →- 1 + g ( x ) = MATH0201 BASIC CALCULUS Limits at a Point Evaluation of Limits Theorem 8 (Algebraic Properties of Limits (p.66)) Let f and g be functions, and assume that lim x → c f ( x ) = L and lim x → c g ( x ) = M ....

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