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Mobile App Chapter 9: Problem 759
The value of \(\lim _{\mathrm{x} \rightarrow(\pi / 4)}\left[\left\\{(\tan \mathrm{x})^{\tan \mathrm{x}}-\tan \mathrm{x}\right\\} /\\{\ln (\tan \mathrm{x})-\tan \mathrm{x}+1\\}\right]\) is (a) 0 (b) 1 (c) \(-2\) (d) none of these
Short Answer
Expert verified
The value of the limit as x approaches \(\pi/4\) is \(0\). (a)
Step by step solution
01
Rewrite the expression in a convenient form
To begin with, we'll rewrite the expression in a more convenient form, using the fractions and exponents defined in the question. The expression can be rewritten as:
\( \lim _{x\to(\pi /4)} \frac{[\tan^{\tan x} x - \tan x]}{\ln(\tan x) - \tan x +1}\)
02
Evaluate the limits of each function involved
First, let's find the limits of each function as x approaches \(\pi/4\):
\(\lim_{x\to(\pi /4)} \tan x = 1\)
\(\lim_{x\to(\pi /4)} \tan^{\tan x} x = 1^{1} = 1\)
\(\lim_{x\to(\pi /4)} \ln(\tan x) = \ln(1) = 0\)
03
Substitute the limit values and check for indeterminate forms
Now, substitute the limit values in the expression:
\(\lim _{x\to(\pi /4)} \frac{[\tan^{\tan x} x - \tan x]}{\ln(\tan x) - \tan x +1} = \frac{[1 -1]}{0-1+1}\)
At this point, we could encounter an indeterminate form. However, in this case, we got a determinate form since we have:
\(\frac{[1-1]}{0-1+1} = \frac{0}{0}\)
This is the indeterminate form of 0/0, so we need to use L'Hopital's rule to find the limit.
04
Apply L'Hopital's Rule
Differentiate the numerator and the denominator.
Numerator derivative:
\(\frac{d(\tan^{\tan x}x - \tan x)}{dx} = \frac{d(\tan^{\tan x}x)}{dx} - \frac{d(\tan x)}{dx}\)
Using the chain rule for \(\tan^{\tan x}x\), we have:
\(\left(\tan^{\tan x}x \right)' = (\tan x)^{(\tan x - 1)}(\tan x)' (\tan x + \log(\tan x))\)
Denominator derivative:
\(\frac{d(\ln(\tan x) - \tan x + 1)}{dx} = \frac{d(\ln(\tan x))}{dx} - \frac{d(\tan x)}{dx} + 0\)
Using the chain rule for \(\ln({\tan x})\), we have:
\(\left(\ln(\tan x) \right)' = (\tan x)'/\tan x\)
Now, let's apply L'Hopital's Rule:
\(\lim_{x\to(\pi /4)} \frac{[\tan^{\tan x} x - \tan x]'}{[\ln(\tan x) - \tan x +1]'} = \frac{(\tan x)^{(\tan(x-1))}(\tan x)' (\tan x + \ln(\tan x)) - (\tan x)'}{(\tan x)'/\tan x}\)
Notice that both, the numerator and the denominator have the term \((\tan x)'\). We can remove this common term from both, giving:
\(\lim_{x\to(\pi /4)} \frac{(\tan x)^{(\tan(x-1))} (\tan x + \ln(\tan x)) - 1}{1/\tan x}\)
05
Evaluate the limit
Evaluate the limit as x approaches \(\pi/4\):
\(\frac{(\tan(\pi/4))^{(\tan(\pi/4)-1)}(\tan(\pi/4) + \ln(\tan(\pi/4))) - 1}{1/\tan(\pi/4)}=\frac{1^{(1-1)}(1 + 0) - 1}{1} = \frac{1-1}{1} = 0\)
The limit of the given expression as x approaches \(\pi/4\) is 0.
So, the answer is (a) 0.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
L'Hopital's Rule
In the world of calculus, L'Hopital's Rule is a lifesaver when it comes to indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). This rule states that if you have a limit that results in an indeterminate form, you can find the limit by taking the derivative of the numerator and the derivative of the denominator and then evaluating the limit of that new fraction.
It's like simplifying a complex expression step by step. If you differentiate the top and bottom separately and still get an indeterminate form, don't fret! Just apply L'Hopital's Rule again. Continue this process until you reach a determinate form or until it becomes evident that the limit does not exist. Remember, this rule only applies when the original limit leads to \(0/0\) or \(\infty/\infty\), and when the derivatives exist and are continuous near the point of interest.
It's like simplifying a complex expression step by step. If you differentiate the top and bottom separately and still get an indeterminate form, don't fret! Just apply L'Hopital's Rule again. Continue this process until you reach a determinate form or until it becomes evident that the limit does not exist. Remember, this rule only applies when the original limit leads to \(0/0\) or \(\infty/\infty\), and when the derivatives exist and are continuous near the point of interest.
Indeterminate Forms
Indeterminate forms are like riddles in calculus; they're expressions that don't give us enough information on their own to decide the limit's value. The most common indeterminate forms include \(0/0\), \(\infty/\infty\), \(0\cdot\infty\), \(\infty - \infty\), \(0^0\), \(\infty^0\), and \(1^\infty\).
When you encounter these in a limit, it suggests that you need a more in-depth analysis to find the limit's value since plugging in numbers directly won't work. Techniques like factoring, conjugation, or using L'Hopital's Rule mentioned earlier, are potential tools to resolve these puzzling expressions. Indeterminate forms are like doors that require special keys; once you have the right method, you unlock the limit's value.
When you encounter these in a limit, it suggests that you need a more in-depth analysis to find the limit's value since plugging in numbers directly won't work. Techniques like factoring, conjugation, or using L'Hopital's Rule mentioned earlier, are potential tools to resolve these puzzling expressions. Indeterminate forms are like doors that require special keys; once you have the right method, you unlock the limit's value.
Tangent Function Limits
Tackling limits involving the tangent function can feel like walking a tightrope; you need to balance carefully to avoid falling into undefined territory. As the tangent function, \(\tan(x)\), combines the sine and cosine functions, its limits can become complex, especially around points where \(\cos(x)\) is zero, as the tangent function heads off to infinity.
But in our exercise, we consider the limit as x approaches \(\pi/4\), where \(\tan(\pi/4)\) is nicely equal to 1. However, caution is warranted as this tame behavior is not a guarantee for all x-values, particularly near the odd multiples of \(\pi/2\), where the function exhibits asymptotic behavior. Understanding the periodic and unbounded nature of the tangent function is vital when examining its limits.
But in our exercise, we consider the limit as x approaches \(\pi/4\), where \(\tan(\pi/4)\) is nicely equal to 1. However, caution is warranted as this tame behavior is not a guarantee for all x-values, particularly near the odd multiples of \(\pi/2\), where the function exhibits asymptotic behavior. Understanding the periodic and unbounded nature of the tangent function is vital when examining its limits.
