Question

In Exercises 2.4.2-2.4.40, find the indicated limits. $$ \lim _{x \rightarrow 0+}\left[\frac{1}{x}+\log (\tan x)\right] $$

Short answer

Answer: The limit of the expression $$\frac{1}{x}+\log (\tan x)$$ as $$x$$ approaches 0 from the right does not exist (DNE).

Step-by-step solution

Analyze the $$\frac{1}{x}$$ function

As x approaches 0 from the right, the function $$\frac{1}{x}$$ approaches positive infinity. This is because the denominator, x, is getting smaller and smaller when we approach 0 from the right, causing the fraction to increase without bound.

Analyze the $$\log(\tan x)$$ function

When analyzing the $$\log(\tan x)$$ part, we can use the property of logarithms that states $$\log ab = \log a + \log b$$ to rewrite the expression as: $$\log(\sin x) + \log(\cos x)^{-1}$$ As x approaches 0 from the right, the sine function will approach 0, and the cosine function will approach 1. Therefore, the logarithmic expressions can be evaluated as: $$\lim _{x \rightarrow 0+}(\log(\sin x)) + \lim _{x \rightarrow 0+} (\log(\cos x)^{-1}) = \log(0) + \log(1) = -\infty + 0 = -\infty$$ Now that we have analyzed both parts of the expression, we can move to the next step.

Combine the two parts to find the limit of the entire expression

The original limit is: $$\lim _{x \rightarrow 0+}\left[\frac{1}{x}+\log (\tan x)\right]$$ We know from Steps 1 and 2 that, the limit of the two parts of the expression as x approaches 0 from the right is: $$\lim_{x \rightarrow 0+}\frac{1}{x} = +\infty$$ and $$\lim_{x \rightarrow 0+}\log (\tan x) = -\infty$$ When adding these two together, the limit does not converge to a finite value. Therefore, the limit of the entire expression does not exist: $$\lim _{x \rightarrow 0+}\left[\frac{1}{x}+\log (\tan x)\right] = DNE$$ In this case, DNE stands for "does not exist."

Key concepts

Infinite Limits

The concept of infinite limits is crucial when we explore functions whose output grows without bound as the input approaches a certain value. Consider the limit \(\lim_{x \rightarrow 0+}\frac{1}{x}\), as found in our exercise. As x gets closer to zero from the positive side, the value of \(\frac{1}{x}\) grows larger and larger, implying that it is unbounded. This unbounded growth signifies that the limit approaches positive infinity, denoted by \(+\infty\).

In graphical terms, the curve of the function would steeply rise towards the vertical line (asymptote) x = 0. The concept of an asymptote is important here: it is a line that the graph of a function approaches but never touches. Infinite limits often indicate the presence of such asymptotes.

Understanding infinite limits can be challenging, but remember these key points: they describe unbounded growth, they can be positive or negative \(\infty\ or \-\infty\), and they are associated with vertical asymptotes in the graph of the function.

Logarithmic Functions

Logarithmic functions play a pivotal role in mathematics, particularly in calculus. The logarithmic function \(\log(x)\), for example, is the inverse of the exponential function and has distinctive properties that make it useful in solving limits. In the given exercise, we encounter the logarithm of a trigonometric function, \(\log(\tan x)\).

To better grasp logarithmic functions, it's essential to understand properties such as \(\log(ab) = \log(a) + \log(b)\), which allows us to break down complex expressions into simpler elements that are easier to evaluate. Moreover, knowing the behavior of the logarithm near zero is key: \(\log(x)\) approaches \-\infty\ as x approaches 0+ because \(\log(1) = 0\) and the function undulates to lower values for inputs between 0 and 1.

Characteristics of Logarithmic Functions

• They have a vertical asymptote at x = 0.
• Their output decreases without bound as x approaches 0 from the right.
• They increase as x increases beyond 1.

These characteristics are essential when evaluating the limits of logarithmic functions.

Trigonometric Limits

Trigonometric limits involve the limits of functions such as sine, cosine, tangent, and their inverses. These types of limits are prevalent in calculus and can be challenging due to the oscillatory nature of trigonometric functions. In our exercise, the trigonometric limit is \(\lim_{x \rightarrow 0+}\log(\tan x)\).

As x approaches 0 from the right, the Tan function \(\tan x\) approaches 0, reflecting the behavior of Sine divided by Cosine, both evaluated at x. For small angles, Sine is approximately equal to the angle (in radians), and Cosine is approximately 1. This makes the tangent of a small positive x also small and positive. However, the logarithm of a small positive number is negative and becomes more negative as the number gets closer to zero.

Remember these key points about trigonometric limits:

• They often require the use of trigonometric identities to be solved.
• Understanding the behavior of trigonometric functions around specific points such as 0, \(\frac{\pi}{2}\), and \(\pi\) is fundamental.
• L'Hôpital's Rule can sometimes aid in computing tricky trigonometric limits.

By combining knowledge of logarithms with the properties of trigonometric functions, we can tackle complex limits involving both fields of mathematics.