Question

Evaluate the following limits or state that they do not exist. $$\lim _{x \rightarrow 0^{+}} \frac{x}{\ln x}$$

Short answer

Answer: The limit of the function $$\frac{x}{\ln x}$$ as $$x$$ approaches 0 from the positive side is 0.

Step-by-step solution

Identify the limit under evaluation

We are asked to find the limit of the function $$f(x) = \frac{x}{\ln x}$$ as $$x$$ approaches 0 from the positive side. It is important to note that this is expressed as: $$\lim_{x \rightarrow 0^+} \frac{x}{\ln x}$$

Analyze the function as x approaches 0 from the positive side

As $$x$$ approaches 0 from the positive side, $$x$$ itself approaches to 0, and also, the natural logarithm of 0 is negative infinity ($$\ln 0 = -\infty$$). Therefore, the limit has the indeterminate form of the type $$\frac{0}{-\infty}$$.

Apply L'Hôpital's Rule

Since the function has an indeterminate form, we can apply L'Hôpital's Rule, which states that if the limit has the indeterminate form of $$\frac{0}{0}$$ or $$\frac{\pm \infty}{\pm \infty}$$, then the limit can be evaluated as follows: $$\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}$$ provided the limit of the right side exists or is infinite. In our case, $$f(x) = x$$ and $$g(x) = \ln x$$. We find their derivatives: $$f'(x) = \frac{d}{dx} x = 1$$, $$g'(x) = \frac{d}{dx} \ln x = \frac{1}{x}$$. Now, let's apply L'Hôpital's Rule: $$\lim_{x \rightarrow 0^+} \frac{x}{\ln x} = \lim_{x \rightarrow 0^+} \frac{f'(x)}{g'(x)} = \lim_{x \rightarrow 0^+} \frac{1}{\frac{1}{x}} = \lim_{x \rightarrow 0^+} x$$

Evaluate the limit

Now that we have simplified the limit, we can evaluate it directly: $$\lim_{x \rightarrow 0^+} x = 0$$ So, the final answer is: $$\lim _{x \rightarrow 0^{+}} \frac{x}{\ln x} = 0$$

Key concepts

L'Hôpital's Rule

Understanding L'Hôpital's Rule can be incredibly helpful when dealing with tricky limits in calculus, especially when you're faced with an indeterminate form. It's named after the French mathematician Guillaume de l'Hôpital, who published the rule in his book in the 17th century.

L'Hôpital's Rule is a strategy to evaluate limits of ratios where both the numerator and the denominator approach either 0 or fty independently. The genius of L'Hôpital's Rule lies in its simplicity: if you have a limit that gives an indeterminate form of frac{0}{0} or frac{±fty}{±fty}, you can instead take the derivative of the top function and the derivative of the bottom function, then find the limit of this new ratio.

Applying the Rule

Always check that the original limit indeed gives an indeterminate form before applying L'Hôpital's Rule. Once verified, differentiate both the top (numerator) and bottom (denominator) functions separately. Then, take the limit of this new ratio of derivatives. If the new limit is determinate, you've found your answer. If it's still indeterminate, you may apply L'Hôpital's Rule again, as long as the conditions for applying the rule are met.

Indeterminate Form

In calculus, you might think that dividing zero by zero or infinity by infinity would give a clear-cut answer, but in reality, these scenarios create what's known as an indeterminate form. This uncertainty occurs because there are actually many possible values these limits could approach, depending on the behavior of the functions involved.

Let's take a closer look: '0/0' can vary greatly. For instance, if the top and bottom functions are approaching zero, but the top does so at a faster rate, the limit might be zero. Conversely, if the bottom function approaches zero faster, the result could be infinity. There are yet other situations where the limit will fall somewhere in between.

Common Indeterminate Forms

Aside from '0/0' and '∞/∞', other indeterminate forms include '0·∞', '∞ - ∞', '1^∞', '0^0', and '∞^0'. Special methods, like L'Hôpital's Rule, can often be employed to find the limits in these cases.

Natural Logarithm

The natural logarithm is a fundamentally important mathematical function, particularly in the world of calculus. It is denoted as 'ln' and represents the logarithm to the base 'e', where 'e' is an irrational and transcendental number approximately equal to 2.71828. This constant 'e' arises in various areas of mathematics, especially in relation to growth problems or processes involving continuous compounding.

The natural logarithm of a number is the power to which 'e' must be raised to produce that number. For example, if we have 'ln(x)', we are essentially asking 'to what power do we need to raise 'e' to get 'x'?'. When 'x' approaches zero, 'ln(x)' approaches negative infinity, which plays a significant role in determining the behavior of limits in calculus.

Derivatives

In calculus, derivatives are all about instantaneous rates of change. Think of them as a snapshot of how a function is behaving at any given moment. If you have a curve that represents a function, the derivative at a point gives you the slope of the tangent to that curve at that point.

Taking derivatives involves rules that enable you to systematically find this rate of change no matter how complex a function is. Some of these rules include the power rule, product rule, quotient rule, and chain rule. When applying L'Hôpital's Rule in limit problems, we use derivatives to transform an indeterminate form into a form that we can deal with more easily. Being able to find a derivative is a powerful tool in the mathematician's arsenal, allowing us to analyze motion, optimization problems, and much more within a variety of scientific fields.