# (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}

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