Handouts3 - Limits - MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Limits at a Point Twosided and Onesided Limits MATH0201 BASIC CALCULUS Limits

# Handouts3 - Limits - MATH0201 BASIC CALCULUS MATH0201 BASIC...

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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 0 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 0 f ( x ) = 0. lim x 0 - x = lim x 0 + f ( x ) = 0 .
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 0 f ( x ) DOES NOT exist. lim x 0 - | x | x = - 1 and lim x 0 + | 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 ) . 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 . 1. For any real k, lim x c k = k; and lim x c x = c. 2. lim x c [ f ( x ) ± g ( x )] = L ± M. 3. For any real number k, lim x c kf ( x ) = kL. 4. lim x c f ( x ) g ( x ) = LM. 5. If M 6 L M .