Problem 21 $$ \lim _{x \rightarrow 0} \op... [FREE SOLUTION]

Chapter 2: Problem 21

$$ \lim _{x \rightarrow 0} \operatorname{sgn}\\{\ln (\cos x)\\}\\{\text { Ans. }-1\\} $$

Short Answer

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The given limit is $\lim_{x\to 0} \operatorname{sgn}(\ln(\cos x))$. As $x$ approaches 0, $\cos(x)$ approaches 1. Taking the natural logarithm, we have $\ln(\cos(0)) = \ln(1) = 0$. Applying the sign function, we get $\operatorname{sgn}(0) = 0$. Therefore, the final answer is $\lim_{x \rightarrow 0} \operatorname{sgn}(\ln(\cos x)) = 0$.

Step by step solution

The sign function, $\operatorname{sgn}(x)$, returns the sign of a given number $x$. Specifically: -$\operatorname{sgn}(x) = 1$, if $x > 0$ -$\operatorname{sgn}(x) = 0$, if $x = 0$ -$\operatorname{sgn}(x) = -1$, if $x < 0$ In this case, we will need to find the value of $\ln (\cos x)$ when $x \rightarrow 0$ and then apply the sign function on that value. #Step 2: The cosine function at x = 0#

The cosine function, $\cos(x)$, oscillates between -1 and 1 as the input $x$ varies. We need to consider its behavior as $x$ approaches 0. We recall that $\cos(0) = 1$, meaning that as $x$ approaches 0, $\cos(x)$ approaches 1. #Step 3: The natural logarithm function#

As we found in the previous step that $\cos(x)$ approaches 1 as $x \rightarrow 0$, we can now take the natural logarithm of this limiting value. We recall the properties of the natural logarithm function, $\ln$: -$\ln(1) = 0$ -$\ln(x)$ is defined only for $x > 0$ Since $\cos(x)$ approaches 1 as $x \rightarrow 0$, we can now find the limiting value for $\ln(\cos x)$ as $x \rightarrow 0$. We get: $\lim_{x \rightarrow 0} \ln(\cos x) = \ln(\cos(0)) = \ln(1)=0$ #Step 4: Apply the sign function#

Now that we have found the limiting value of $\ln(\cos x)$ as $x \rightarrow 0$, which is 0, we can apply the sign function to it: $\lim_{x \rightarrow 0} \operatorname{sgn}(\ln(\cos x)) = \operatorname{sgn}(0)$ Using the definition of the sign function that we found in Step 1, we have: $\operatorname{sgn}(0) = 0$ Therefore, the final answer is: $\lim_{x \rightarrow 0} \operatorname{sgn}(\ln(\cos x)) = 0$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sign Function

The sign function, represented as $\operatorname{sgn}(x)$, is a very simple yet powerful concept in mathematics. It tells us whether a number is positive, negative, or zero. Here's how it works:

If $x > 0$, then $\operatorname{sgn}(x) = 1$.
If $x = 0$, then $\operatorname{sgn}(x) = 0$.
If $x < 0$, then $\operatorname{sgn}(x) = -1$.

This function is particularly important in calculus when dealing with limits and function behavior. For example, in the given problem, once we determine the behavior of $\ln(\cos x)$ as $x$ approaches 0, the sign function helps us determine the nature of the result.

Natural Logarithm

Natural logarithms, denoted as $\ln(x)$, are logarithms with base $e$, where $e$ is an irrational and transcendental constant approximately equal to 2.71828. The natural logarithm function is fundamental in calculus due to its natural occurrence in various growth processes and its properties in differentiation. Here are some key points to remember:

$\ln(1) = 0$
$\ln(x)$ is undefined for $x \leq 0$, making it important to ensure that the input is positive.
The function is continuous and differentiable for all positive $x$.

In our exercise, we're interested in $\ln(\cos x)$ as $x$ approaches 0. Since $\cos(x)$ approaches 1 in this scenario, the logarithmic value simplifies to 0, as $\ln(1) = 0$. This simplification makes subsequent calculations straightforward.

Trigonometric Limits

Trigonometric limits are crucial in calculus when evaluating the behavior of trigonometric functions as their inputs approach a certain value. These functions, such as $\sin(x)$, $\cos(x)$, and $\tan(x)$, often appear in limit problems. Here鈥檚 a quick overview:

$\lim_{x \rightarrow 0} \sin(x) = 0$
$\lim_{x \rightarrow 0} \cos(x) = 1$
$\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1$

In our problem, the focus is on $\cos(x)$ as $x$ heads towards 0. Since $\cos(x)$ tends towards 1, any function depending on this limit, like $\ln(\cos x)$, simplifies to involve constants. Understanding such basic limits helps significantly in more complex calculus problem-solving.

Calculus Problem Solving

Calculus problem solving often involves a systematic approach to evaluate limits, derivatives, or integrals. Here are a few steps that can streamline this process:

Understand the Problem: Identify what is being sought. In this problem, it's the limit of the sign of a natural logarithm function.
Analyze Component Functions: Break down the problem, understanding how individual functions like $\cos(x)$ and $\ln(x)$ behave as $x$ approaches specific values.
Solve Sequentially: Tackle simpler parts first and use those results in more complex computations. For instance, find $\ln(\cos x)$ before applying the sign function.
Use Known Limits and Theorems: Incorporate known limits like $\cos(x) \to 1$ as $x \to 0$ to simplify calculations.

By structuring your approach and using these strategies, challenging calculus questions can become more manageable. This process aids not just in finding answers, but also in building a deeper understanding of the function behaviors underlying the problem.

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$$ \lim _{x \rightarrow 0} \operatorname{sgn}\\{\ln (\cos x)\\}\\{\text { Ans. }-1\\} $$

Short Answer

Step by step solution

Key Concepts

Sign Function

Natural Logarithm

Trigonometric Limits

Calculus Problem Solving

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