Chapter 4 | Applications of Derivatives
461
Figure 4.72 Finding the limit at x =0 of the function f ( x ) = x ln x .
Evaluate lim x →0
x cot x .
4.40
Indeterminate Form of Type ∞−∞ Another type of indeterminate form is ∞−∞. Consider the following example. Let n be a positive integer and let f ( x ) =3 x n and g ( x ) =3 x 2 +5. As x →∞, f ( x )→∞ and g ( x )→∞. We are interested in lim x →∞ ⎛ ⎝ f ( x )− g ( x ) ⎞ ⎠ . Depending on whether f ( x ) grows faster, g ( x ) grows faster, or they grow at the same rate, as we see next, anything can happen in this limit. Since f ( x )→∞ and g ( x )→∞, we write ∞−∞ to denote the form of this limit. As with our other indeterminate forms, ∞−∞ has no meaning on its own and we must do more analysis to determine the value of the limit. For example, suppose the exponent n in the function f ( x ) =3 x n is n =3, then lim x →∞ ⎛ ⎝ f ( x )− g ( x ) ⎞ ⎠ = lim x →∞ ⎛ ⎝ 3 x 3 −3 x 2 −5 ⎞ ⎠ =∞. On the other hand, if n =2, then lim x →∞ ⎛ ⎝ f ( x )− g ( x ) ⎞ ⎠ = lim x →∞ ⎛ ⎝ 3 x 2 −3 x 2 −5 ⎞ ⎠ =−5. However, if n =1, then lim x →∞ ⎛ ⎝ f ( x )− g ( x ) ⎞ ⎠ = lim x →∞ ⎛ ⎝ 3 x −3 x 2 −5 ⎞ ⎠ =−∞. Therefore, the limit cannot be determined by considering only ∞−∞. Next we see how to rewrite an expression involving the indeterminate form ∞−∞ as a fraction to apply L’Hôpital’s rule. Example 4.42 Indeterminate Form of Type ∞−∞
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