Problem 42
Question
Use numerical or graphical means to find the limit, if it exists. If the limit of f as x approaches c does exist, answer this question: Is it equal to \(f(c) ?\) $$\lim _{x \rightarrow \pi / 2} x \sin x$$
Step-by-Step Solution
Verified Answer
Answer: The limit of the function \(x\sin x\) as \(x\) approaches \(\pi/2\) is \(\frac{\pi}{2}\), and yes, this limit is equal to the value of the function at \(x=\pi/2\).
1Step 1: Analytically find the limit
We are given the function \(x\sin x\) and we are asked to find its limit as \(x\) approaches \(\pi/2\). To do this analytically, we will simply evaluate the function for the given value of \(x\).
$$\lim _{x \rightarrow \pi / 2} x \sin x = \frac{\pi}{2} \sin \frac{\pi}{2}$$
Since \(\sin \frac{\pi}{2} = 1\), the limit is:
$$\lim _{x \rightarrow \pi / 2} x \sin x = \frac{\pi}{2}$$
2Step 2: Compare the limit with the function's value
Now that we have the limit, we need to compare it with the value of the function at \(x=\pi/2\).
The function is defined for \(x=\pi/2\), since there is no discontinuity at this point. So, we can find the value of the function at \(x=\pi/2\) as follows:
$$f(\pi / 2) = \frac{\pi}{2} \sin \frac{\pi}{2} = \frac{\pi}{2}$$
Since the limit and the value of the function are equal:
$$\lim _{x \rightarrow \pi / 2} x \sin x = f(\pi / 2)$$
So the answer to the question "Is the limit equal to \(f(\pi/2)\)" is yes.
Key Concepts
Limits of Trigonometric FunctionsEvaluating Limits AnalyticallyContinuity of Functions at a Point
Limits of Trigonometric Functions
Trigonometric functions like sine, cosine, and tangent often appear in calculus problems, and their limits can initially be puzzling due to their oscillatory nature. However, their limits can usually be found using direct substitution, as long as the functions are not approaching an undefined value. For example, in the provided exercise, we looked at the limit of the product of \(x\) and \(\sin x\) as \(x\) approaches \(\pi/2\).
The limit was found through direct evaluation after confirming that \(\sin(\pi/2)\) is indeed defined and equal to 1. Therefore, we computed the limit as \(\frac{\pi}{2}\) by simply substituting \(\pi/2\) into the function. While not all limits of trigonometric functions are this straightforward—especially when they involve angles that approach vertical asymptotes or result in forms like \(0/0\)—this example shows how a direct substitution can be effective in certain cases.
Remember to consider the domain of the trigonometric functions and be mindful of angles that cause the function to be undefined. In cases of indeterminate forms, other techniques such as L'Hôpital's Rule may be necessary to evaluate the limit.
The limit was found through direct evaluation after confirming that \(\sin(\pi/2)\) is indeed defined and equal to 1. Therefore, we computed the limit as \(\frac{\pi}{2}\) by simply substituting \(\pi/2\) into the function. While not all limits of trigonometric functions are this straightforward—especially when they involve angles that approach vertical asymptotes or result in forms like \(0/0\)—this example shows how a direct substitution can be effective in certain cases.
Remember to consider the domain of the trigonometric functions and be mindful of angles that cause the function to be undefined. In cases of indeterminate forms, other techniques such as L'Hôpital's Rule may be necessary to evaluate the limit.
Evaluating Limits Analytically
When faced with the challenge of finding a limit, the analytical approach entails a deeper look into the function itself and understanding its behavior as it approaches a certain point. Evaluating limits analytically can be done through various methods, including direct substitution, factoring, conjugation, and the use of L'Hôpital's Rule. In the given exercise, the technique used was remarkably straightforward: the limit was calculated by directly substituting the value to which \(x\) approaches—in this case, \(\pi/2\).
This is possible when the function is continuous at the point of interest. However, if direct substitution yields an indeterminate form such as \(0/0\) or \(\infty/\infty\), other analytical methods must be employed. For instance, factoring might simplify the expression to a form that allows direct substitution, or L'Hôpital's Rule can resolve indeterminate forms by differentiating the function's numerator and denominator separately. Each scenario demands a tailored approach to evaluating the limit analytically.
This is possible when the function is continuous at the point of interest. However, if direct substitution yields an indeterminate form such as \(0/0\) or \(\infty/\infty\), other analytical methods must be employed. For instance, factoring might simplify the expression to a form that allows direct substitution, or L'Hôpital's Rule can resolve indeterminate forms by differentiating the function's numerator and denominator separately. Each scenario demands a tailored approach to evaluating the limit analytically.
Continuity of Functions at a Point
The concept of continuity at a point is pivotal in understanding function behavior and is closely tied with limits. A function is said to be continuous at a point \(x = c\) if three conditions are met: the function is defined at \(c\), the limit of the function as \(x\) approaches \(c\) exists, and the limit is equal to the function's value at \(c\).
In the context of our example, we not only calculated the limit as \(x\) approaches \(\pi/2\) but also confirmed that the function's value at \(x = \pi/2\) is the same as the limit. This shows continuity at \(\pi/2\) because it satisfies all three conditions. Ensuring that a function is continuous at a point is essential, especially when dealing with the Fundamental Theorem of Calculus or when determining whether a particular value can be used in the function without causing disruption, such as a jump or an asymptote.
Understanding whether a function is continuous at a particular point needs a good grasp of limits and the behavior of the function around that point. This concept is key in connecting the dots between the limit of a function and its pointwise evaluation.
In the context of our example, we not only calculated the limit as \(x\) approaches \(\pi/2\) but also confirmed that the function's value at \(x = \pi/2\) is the same as the limit. This shows continuity at \(\pi/2\) because it satisfies all three conditions. Ensuring that a function is continuous at a point is essential, especially when dealing with the Fundamental Theorem of Calculus or when determining whether a particular value can be used in the function without causing disruption, such as a jump or an asymptote.
Understanding whether a function is continuous at a particular point needs a good grasp of limits and the behavior of the function around that point. This concept is key in connecting the dots between the limit of a function and its pointwise evaluation.
Other exercises in this chapter
Problem 41
Find $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=x^{2}$$
View solution Problem 42
A function \(f\) that is not defined at \(x=c\) is said to have \(a\) removable discontinuity at \(x=c\) if there is a function \(g\) such that \(g(c)\) is defi
View solution Problem 42
Find $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=x^{3}$$
View solution Problem 43
Find the limit. $$\lim _{x \rightarrow \infty} \frac{x}{|x|}$$
View solution