# 0114 - Math 31A 2010.01.14 MATH 31A DISCUSSION JED YANG 1...

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Unformatted text preview: Math 31A 2010.01.14 MATH 31A DISCUSSION JED YANG 1. More Limits 1.1. Exercise 2.3.30. Show that the Product Law cannot be used to evaluate lim x → π/ 2 ( x − π/ 2) tan x . Solution. The Product Law requires the limit of each factor to exist. However, lim x → π/ 2 tan x does not exist. Recall that tan x = sin x cos x . Thus lim x → π/ 2- = + ∞ and lim x → π/ 2+ = −∞ , and either of these imply the limit does not exist. square 1.2. Exercise 2.4.31. Determine the points at which the function f ( x ) = tan(sin x ) is discontinuous and state the type of discontinuity: removable, jump, infinite, or none of these. Solution. Recall that tan x is discontinuous at x = kπ/ 2 for odd k . However, we have − 1 ≤ sin x ≤ 1, and tan x is continuous on [ − 1 , 1], so tan(sin x ) is continuous everywhere. square 1.3. Exercise 2.4.48. Sawtooth Function. Draw the graph of f ( x ) = x − [ x ]....
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0114 - Math 31A 2010.01.14 MATH 31A DISCUSSION JED YANG 1...

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