Question

Evaluate the following limits. $$\lim _{x \rightarrow 0^{+}} \frac{x}{\ln x}$$

Short answer

Answer: The limit of the function as x approaches 0 from the positive side is 0.

Step-by-step solution

Identify the indeterminate form

Before using L'Hôpital's rule, we need to confirm that the expression has an indeterminate form when x approaches 0 from the positive side. We can plug in 0 for x: $$\lim_{x \rightarrow 0^{+}} \frac{x}{\ln x} = \frac{0}{\ln 0}$$ Since the natural logarithm is undefined for 0, this limit is an indeterminate form.

Apply L'Hôpital's rule

L'Hôpital's rule states that, if the limit of the ratio of the derivatives exists, then the limit of the original function is equal to that limit. Therefore, we will find the derivatives of both the numerator (x) and the denominator (ln(x)) with respect to x: Numerator derivative: $$\frac{d}{dx}(x) = 1$$ Denominator derivative: $$\frac{d}{dx}(\ln x) = \frac{1}{x}$$ Now we can take the limit of the ratio of these derivatives: $$\lim_{x \rightarrow 0^{+}} \frac{\frac{d}{dx}(x)}{\frac{d}{dx}(\ln x)} = \lim_{x \rightarrow 0^{+}} \frac{1}{\frac{1}{x}}$$

Evaluate the limit

Now that we have the limit of the ratio of the derivatives, we can plug in 0 for x and see if the limit exists: $$\lim_{x \rightarrow 0^{+}} \frac{1}{\frac{1}{x}} = \frac{1}{\frac{1}{0^+}}$$ As x approaches 0 from the positive side, the fraction in the denominator becomes a very large positive number. Thus, the whole expression approaches 0: $$\lim_{x \rightarrow 0^{+}}\frac{1}{\frac{1}{x}} = 0$$ So, the limit of the given function as x approaches 0 from the positive side is 0. This means that: $$\lim _{x \rightarrow 0^{+}} \frac{x}{\ln x} = 0$$

Key concepts

Limits

Limits are fundamental in calculus and help us understand the behavior of functions. They describe what happens to a function as it approaches a certain point, often resulting in a value or sometimes identifying where the function goes to infinity.

When dealing with limits, it's crucial to analyze what happens as we get close to a particular value, not necessarily at that value itself. In the provided exercise, we examine the limit as \( x \) approaches 0 from the positive side, which we notate as \( x \rightarrow 0^+ \). This implies that we are interested in the behavior of the function just above zero, but never actually reaching zero. Understanding this can help assess situations where direct substitution might not work due to indeterminacies or undefined points.

Indeterminate Forms

Indeterminate forms arise when evaluating limits leads to a mathematical expression that is not immediately clear or defined, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These forms signal an interruption in a straightforward limit calculation, often requiring advanced techniques to resolve.

In the problem, when we approach \( x \rightarrow 0^+ \), the expression \( \lim_{x \rightarrow 0^{+}} \frac{x}{\ln x} \) forms an indeterminate type. Here, the numerator goes to 0, and the natural logarithm in the denominator becomes problematic because \( \ln(0) \) is undefined. To manage such situations, we utilize approaches like L'Hôpital's Rule, which can convert an indeterminate form into a calculable limit.

Derivatives

Derivatives provide the rate of change of a function with respect to one of its variables. They are extremely useful in many areas of calculus, including solving limits involving indeterminate forms. In the exercise, we use derivatives to apply L'Hôpital's rule, a process designed specifically for resolving such forms.

To apply L'Hôpital's Rule, we must find the derivatives of both the numerator, \( x \), and the denominator, \( \ln x \). The derivative of \( x \) is 1, as it represents a linear rate of change. For \( \ln x \), the derivative is \( \frac{1}{x} \), which describes how the natural logarithm function changes as \( x \) varies. By finding these derivatives, we turn the original indeterminate expression into a more manageable form for resolving the limit.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a key function in mathematics, particularly in calculus. It transforms multiplication into addition, which simplifies complex equations significantly. The base of the natural logarithm is the irrational number \( e \), approximately equal to 2.718.

In our problem, the function \( \ln x \) plays a crucial role in forming the indeterminate expression. Natural logarithms are specifically undefined for zero and negative numbers, which is why \( \ln x \) becomes problematic as \( x \) approaches \( 0^+ \). By using the derivative \( \frac{1}{x} \), the changes in \( \ln x \) can be accounted for in limit calculations, turning difficult problems into simpler ones.