Chapter 1: Problem 13
Find the limit. $$ \lim _{x \rightarrow 3} \tan \left(\frac{\pi x}{4}\right) $$
Short Answer
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Step by step solution
01
Understand the Function
The function in question is \(\tan \left(\frac{\pi x}{4}\right)\). We want to find the value of this function as \(x\) approaches 3. The tangent function is a periodic function, but in this interval doesn't have any discontinuities, so we can proceed by direct substitution.
02
Substituting \(x = 3\)
Now, we will substitute 3 into the function \(\tan \left(\frac{\pi x}{4}\right)\), which gives us \(\tan \left(\frac{\pi \cdot 3}{4}\right)\). Thus, we want to determine the value \(\tan \left(\frac{3\pi}{4}\right)\).
03
Evaluating the Limit
\(\tan \left(\frac{3\pi}{4}\right)\) is equivalent to \(-1\),since \(\tan(225º) = \tan(-135º) = -1\), considering the periodicity of the tangent function.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Tangent Function
The tangent function, denoted as \( \tan \(x\) \), is one of the primary trigonometric functions and plays a crucial role in calculus, particularly when dealing with limits, derivatives, and integrals. Understanding its characteristics is essential for solving calculus problems effectively.
The function represents the ratio of the sine and cosine of an angle in a right triangle, and it is defined for all real numbers except for the odd multiples of \( \frac{\pi}{2} \) where the cosine of the angle is zero, causing the function to be undefined. The tangent function is periodic with a period of \( \pi \) and exhibits symmetry, which means \( \tan(\theta + n\pi) = \tan(\theta) \) for any integer \( n \).
For the problem at hand, as \(x\) approaches 3, we are interested in \( \tan \( \left(\frac{\pi x}{4}\right) \) \). The interval from 0 to 3 \( \pi/4 \) doesn't include any of the undefined points, hence we can find the limit without concern for discontinuities. The key is to recognize the tangent's behavior at specific angles, like \( \frac{3\pi}{4} \) in our exercise, to accurately evaluate the limit.
The function represents the ratio of the sine and cosine of an angle in a right triangle, and it is defined for all real numbers except for the odd multiples of \( \frac{\pi}{2} \) where the cosine of the angle is zero, causing the function to be undefined. The tangent function is periodic with a period of \( \pi \) and exhibits symmetry, which means \( \tan(\theta + n\pi) = \tan(\theta) \) for any integer \( n \).
For the problem at hand, as \(x\) approaches 3, we are interested in \( \tan \( \left(\frac{\pi x}{4}\right) \) \). The interval from 0 to 3 \( \pi/4 \) doesn't include any of the undefined points, hence we can find the limit without concern for discontinuities. The key is to recognize the tangent's behavior at specific angles, like \( \frac{3\pi}{4} \) in our exercise, to accurately evaluate the limit.
Calculating the Limit of a Function
The limit of a function is a fundamental concept in calculus, representing the value that a function approaches as the input (or the variable) gets infinitely close to some number. It's a way of describing the behavior of a function at a certain point, usually at points where the function itself may not be well-defined.
When calculating limits, we're typically interested in finding out what value a function is heading towards as we get closer and closer to a particular value of \( x \). In other words, limits help us investigate the function's behavior around a point, even if that point itself is not within the function's domain. For continuous functions, limits can often be calculated simply by substituting the value of \( x \) into the function, known as direct substitution.
However, with functions that are not continuous or exhibit indeterminate forms, we might need to deploy other techniques such as factoring, rationalizing, using trigonometric identities, or applying L'Hôpital's rule. Understanding limits is vital, as they are the building blocks of more advanced concepts like derivatives (which measure the rate of change) and integrals (which calculate the area under a curve).
When calculating limits, we're typically interested in finding out what value a function is heading towards as we get closer and closer to a particular value of \( x \). In other words, limits help us investigate the function's behavior around a point, even if that point itself is not within the function's domain. For continuous functions, limits can often be calculated simply by substituting the value of \( x \) into the function, known as direct substitution.
However, with functions that are not continuous or exhibit indeterminate forms, we might need to deploy other techniques such as factoring, rationalizing, using trigonometric identities, or applying L'Hôpital's rule. Understanding limits is vital, as they are the building blocks of more advanced concepts like derivatives (which measure the rate of change) and integrals (which calculate the area under a curve).
Direct Substitution in Limits
Direct substitution is often the simplest and first method to attempt when finding the limit of a function as \( x \) approaches a particular value. The technique involves replacing the variable \( x \) with the number that it's approaching in the limit expression.
Proper use of direct substitution requires that the function be continuous at the point of interest. Continuity implies that there are no breaks, jumps, or holes in the graph of the function at that point. If the function is not continuous, direct substitution might lead to misleading or incorrect results. In the case of limits that lead to an undefined form like \(0/0\) or \(\infty/\infty\), alternative strategies must be used.
For the given exercise, we are in luck since the function \( \tan \( \left(\frac{\pi x}{4}\right) \) \) is continuous around \(x=3\) and direct substitution can be safely used. By plugging \(3\) into the function, we get \( \tan \( \left(\frac{3\pi}{4}\right) \) \) and can straightaway evaluate the tangent of that angle to find the limit, which, as outlined in the solution steps, is \( -1 \) due to the properties of the tangent function.
Proper use of direct substitution requires that the function be continuous at the point of interest. Continuity implies that there are no breaks, jumps, or holes in the graph of the function at that point. If the function is not continuous, direct substitution might lead to misleading or incorrect results. In the case of limits that lead to an undefined form like \(0/0\) or \(\infty/\infty\), alternative strategies must be used.
For the given exercise, we are in luck since the function \( \tan \( \left(\frac{\pi x}{4}\right) \) \) is continuous around \(x=3\) and direct substitution can be safely used. By plugging \(3\) into the function, we get \( \tan \( \left(\frac{3\pi}{4}\right) \) \) and can straightaway evaluate the tangent of that angle to find the limit, which, as outlined in the solution steps, is \( -1 \) due to the properties of the tangent function.
