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CHAPTER 2

CHAPTER 2. 2.4 Continuity. Indeterminate Forms and L’Hospital’s Rule. L’Hospital’s Rule Suppose f and g are differentiable and g’ ( x )  0 near a . Suppose that

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CHAPTER 2

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  1. CHAPTER 2 2.4 Continuity Indeterminate Forms and L’Hospital’s Rule

  2. L’Hospital’s Rule Suppose f and g are differentiable and g’(x)  0 neara. Suppose that lim x af(x) = 0 and lim x ag(x) = 0, or lim x af(x) = oo and lim x ag(x) = oo Then lim x af(x) / g(x) = lim x af’(x)/g’(x) if the limit on the right side exists.

  3. Example Find limx tan x /x. CHAPTER 2 2.4 Continuity

  4. Example Find lim x ooln (1 + ex ) / 5x. CHAPTER 2 2.4 Continuity

  5. Indeterminate Products If lim x a f(x) = 0 and lim x a g(x) = oo, then it’s not clear what the value of lim x a f(x)g(x). We can find the value of this limit by writing the product fg as a quotient : fg = f /(1 / g) or fg = g / (1 / f ).

  6. Example Find lim x ooxex .

  7. Indeterminate Differences If lim x a f(x) = oo and lim x a g(x) = oo, then the limit lim x a [ f(x) - g(x) ] is called an indeterminate form of type oo - oo.

  8. Example Find lim x 0 ( csc x – cot x ).

  9. Indeterminate Powers Several indeterminate forms arise from lim x a[ f(x) ]g(x) . • lim x af(x) = 0 and lim x ag(x) = 0 (type 00). • lim x af(x) = oo and lim x ag(x) = oo (type oo0 ). • lim x af(x) = 1 and lim x ag(x) =  oo (type 1oo ).

  10. Example Calculate lim x oo( 1 + ( a / x ))bx.

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