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(a) Identify the indeterminate form of lim x ? 0 ? [ 1 x + ln ( ? 1 x ) ] and rewrite it as a...

Question:

(a) Identify the indeterminate form of {eq}\lim_{x \rightarrow 0^-} \left [ \frac{1}{x} + \ln(\frac{-1}{x}) \right ] {/eq} and rewrite it as a quotient without using L'hospital's rule.

(b) Is your quotient in indeterminate form?

(c) Use the answer you got in part (a) to show that the limit does not exist.

Existence of a Limit:

Suppose that {eq}a(x) {/eq} is a function that is defined in an interval that contains {eq}x=t {/eq}. Then the limit is defined as:

{eq}\mathop {\lim }\limits_{x \to t} a(x) = M {/eq}

There exists a very small number {eq}k {/eq} such that {eq}k>0 {/eq} so that {eq}n>0 {/eq}. This means that {eq}|a(x) - M| < k {/eq} whenever {eq}0 < |x - t| < n {/eq}.

if {eq}\mathop {\lim }\limits_{x \to {t^ + }} a(x) = {A_1} {/eq} and {eq}\mathop {\lim }\limits_{x \to {t^ - }} a(x) = {A_2} {/eq}, then;

(1) The limit exists if {eq}{A_1} = {A_2} {/eq}

(2) The limit does not exist if {eq}{A_1} \ne {A_2} {/eq}

Answer and Explanation:

{eq}\displaystyle \eqalign{ & \mathop {\lim }\limits_{x \to {0^ - }} \left[ {\frac{1}{x} + \ln (\frac{{ - 1}}{x})} \right] \cr & a) \cr & \mathop {\lim }\limits_{x \to {0^ - }} \left[ {\frac{1}{x} + \ln \left( { - \frac{1}{x}} \right)} \right]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\infty - \infty } \right) \cr & \mathop {\lim }\limits_{x \to {0^ - }} \left[ {\frac{{1 + x\ln \left( { - \frac{1}{x}} \right)}}{x}} \right] \cr & \cr & b) \cr & {\text{No}} \cr & \mathop {\lim }\limits_{x \to {0^ - }} \left[ {\frac{{1 + x\ln \left( { - \frac{1}{x}} \right)}}{x}} \right]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\frac{1}{0}} \right) \cr & \cr & c) \cr & \mathop {\lim }\limits_{x \to {0^ - }} \left[ {\frac{1}{x} + \ln \left( { - \frac{1}{x}} \right)} \right] \cr & \left[ {\mathop {\lim }\limits_{x \to {0^ - }} \frac{1}{x} + \mathop {\lim }\limits_{x \to {0^ - }} \ln \left( { - \frac{1}{x}} \right)} \right]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{From limit properties}}} \right) \cr & \left[ { - \infty + \infty } \right]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{From limit properties}}} \right) \cr & \infty \cr & \cr & \mathop {\lim }\limits_{x \to {0^ - }} \left[ {\frac{1}{x} + \ln \left( { - \frac{1}{x}} \right)} \right] \to {\text{Diverge}} \cr} {/eq}


Learn more about this topic:

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Understanding the Properties of Limits

from Math 104: Calculus

Chapter 6 / Lesson 5
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