Math_137_Winter_2010_Week_3_Notes

Math_137_Winter_2010_Week_3_Notes - Math 137 Week 3 Notes...

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Math 137 Week 3 Notes – Dr. Paula Smith I. The limit of a function. A. Motivation 1. (High school) algebra studied numbers operated on by functions; (university) calculus studies functions operated on by the limit, differential operator, and integral. 2. In this course, we will study Differential Calculus, concerning derivatives; a derivative at point α can be thought of as the slope of a curve at that point. 3. Finding the slope of a straight line f(x) is easy – take any two distinct points of the line ( α ,f( α )) and (x,f(x)); slope is (f(x) – f( α )) / (x – α ). 4. Slope of a curved line f(x) is confusing – which two points to take? 5. Big idea: the slope of a curve at point ( α ,f( α )) is the slope of the straight line tangent to f at that point. 6. We use limits to determine that slope. B. Intuitive ideas 1. As x a, f(x) L – “as x approaches a (from either side), f(x) approaches the limit L.” 2. We can make f(x) as close to L as we like by taking x to be su ciently close to a (though not equal to a). 3. We can make |f(x) L| as small as we like by taking x so that |x a| is su ciently small but not equal to 0. 4. Notation: lim x a f(x) = L a. Be careful; lim x a f(x) and f(a) are not always the same. One might exist while the other doesn’t. C. One-sided limits 1. It can be easier to think of limits restricted to just one side of a . 2. Notation: a) If x > a, lim x a+ f(x) = L. b) If x < a, lim x a- f(x) = L. c) Note that the sign indicating the side is behind a ; so we might have lim x -4- f(x) = L or lim x -4+ f(x) = L if a = – 4. 3. For lim x a f(x) to exist, lim x a+ f(x) = L must exist, lim x a- f(x) = M must exist, and L must equal M. Then lim x a f(x) = L. D. What can go wrong? 1. jump discontinuity: on the left side of α f may approach the value M, and on the right side of α , f may approach the value N, but M N. Since f doesn’t tend to the same value, the limit of f DNE. 2. Infinite wiggle: f(x) = sin( π /x) at x approaches 0 3. Infinity: f(x) = 1/(x – 1) as x approaches 1 E. Hence for the limit of f to exist at point α , the lefthand limit lim x →α - f(x) must exist (equal a real number L), the righthand limit lim x →α + f(x) must exist, and both must equal the same value
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F. Mathematically precise de nition Lim x a f(x) = L if for every ε > 0 there is a corresponding δ > 0 such that 0 < |x a| < δ implies |f(x) L| < ε . G. Examples 1. Use the precise de nition of limit to prove that lim x 2 (3x 7) = 1. Strategy : First we substitute a = 2, f(x) = 3x 7, L = 1 in the de nition. We need to nd a formula for δ in terms of ε such that 0 < |x 2| < δ implies |(3x 7) ( 1)| < ε . Notice that |(3x 7) ( 1)| = |3x – 6| = 3|x – 2| . If |x 2| < δ , then 3|x – 2| < 3 δ . This will be less than
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Math_137_Winter_2010_Week_3_Notes - Math 137 Week 3 Notes...

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