Calc Chapter 1.docx

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MATH 1502

Chapter 1: Misc topics Section 4.1 Indeterminate form – meaningless expression that describes the behavior of a function near a point or as it approaches infinity Ex) 0/0, ∞/∞, 0 * ∞, ∞ - ∞, 0^0, 1^∞, ∞^0 L’Hopital’s Rule: Suppose f(a) = g(a) = 0 and f, g are differentiable on an open interval containing a. Suppose g’(x) is not equal to 0 on this interval (x is not equal to a) Then lim xa f(x)/g(x) = lim xa f’(x)/g’(x) Remarks: 1) differentiate numerator and denominator separately(not quotient rule) 2)keep applying the rule as long as 0/0 remains 3)Be smart ex lim x0 = x^2/x = lim x0 2x/1 = 0 4)everything’s the same for 1 sided limits Ex lim x0+ (zero from the right) x-sin(x)/x^3 = lim x0+ 1-cos(x) =lim x0+ -sin(x)/6x = lim x-> 0+ -cos(x)/6 = 1/6 L’Hopital’s rule works the same for ∞/∞ as it does for 0/0 Ex) lim x∞ ln(x)/(2√x) = lim x∞ (1/x)/(1/√x) = lim x∞ (√x)/x = lim x∞ 1/√x = 0 Sometimes you need to exponentiate Ex) limx0+ = (ln(x) – ln(sin(x)) = lim x+ ln(x/sin(x)) =
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