Problem 39

Question

Test the following functions for continuity $f(x)=\frac{\cos (\ln x)+\sin ^{3} x \cdot \tan ^{-1} x}{e^{x}+\cosh x}$.

Step-by-Step Solution

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Answer

The given function $f(x)=\frac{\cos (\ln x)+\sin ^{3} x \cdot \tan ^{-1} x}{e^{x}+\cosh x}$ is continuous in its domain ($x \in (0, \infty)$), as all components are continuous elementary functions in their respective domains and their compositions, products, and sum are also continuous.

1Step 1: Identify the components

Notice that the given function consists of components that are compositions, products, or sum of elementary functions: $\cos(\ln x)$, $\sin^3(x)$, $\tan^{-1}(x)$, $e^x$, and $\cosh(x)$. All these elementary functions are continuous in their respective domains.

2Step 2: Determine the domain of each component

The domain of each elementary function is as follows: 1. $\cos(\ln x)$: $x \in (0, \infty)$ 2. $\sin^3(x)$: $x \in \mathbb{R}$ 3. $\tan^{-1}(x)$: $x \in \mathbb{R}$ 4. $e^x$: $x \in \mathbb{R}$ 5. $\cosh(x)$: $x \in \mathbb{R}$

3Step 3: Determine the domain of the given function

The domain of the given function is the intersection of the domains of all its components, which is: $x \in (0, \infty)$.

4Step 4: Study the limit of the composite and product functions

Since the elementary functions are continuous in their domains, their compositions and products will also be continuous in their respective domains. So, we have the continuity of $\cos(\ln x)$ on $(0, \infty)$, $\sin^3(x) \cdot \tan^{-1}(x)$ on $\mathbb{R}$, and $e^x + \cosh(x)$ on $\mathbb{R}$.

5Step 5: Study the continuity of the given function

Since $\cos(\ln x)$ and $\sin^3(x) \cdot\tan^{-1}(x)$ are continuous in their domains, their sum is also continuous in the domain of the given function ($x \in (0, \infty)$). Additionally, since $e^x + \cosh(x)$ is continuous and nonzero on $\mathbb{R}$, it is also continuous on the domain of the given function ($x \in (0, \infty)$). Thus, the quotient of these continuous functions, which is the given function $f(x)$, will also be continuous in its domain ($x \in (0, \infty)$). Therefore, the given function $f(x)=\frac{\cos (\ln x)+\sin ^{3} x \cdot \tan ^{-1} x}{e^{x}+\cosh x}$ is continuous in its domain ($x \in (0, \infty)$).

Key Concepts

Elementary FunctionsDomain of FunctionsComposite FunctionsProduct of Functions

Elementary Functions

Elementary functions are the building blocks of more complex mathematical expressions. These include algebraic functions, such as polynomials, and transcendental functions, like trigonometric and exponential functions. In the given exercise, the function is comprised of several elementary functions:

$\cos(\ln x)$ combines logarithmic and trigonometric functions.
$\sin^3(x)$ uses the sine function, raised to the third power.
$\tan^{-1}(x)$ involves the inverse tangent function.
$e^x$ is a classic exponential function.
$\cosh(x)$ is the hyperbolic cosine function.

These functions are essential because they are universally defined and continuous in their respective natural domains, making them reliable when constructing more complex functions.

Domain of Functions

The domain of a function refers to the set of input values (usually $x$) for which the function is defined. Determining the domain is crucial as it helps identify where the function can operate without mathematical errors or undefined behavior.
For the function in the exercise, consider:

$\cos(\ln x)$ has a domain of $(0, \infty)$ because the natural logarithm $\ln x$ only accepts positive numbers.
$\sin^3(x)$, $\tan^{-1}(x)$, $e^x$, and $\cosh(x)$ are defined for all real numbers, $\mathbb{R}$.

The overall domain of the function becomes $(0, \infty)$ as this is the intersection of all the component functions' domains, ensuring the function is well-defined.

Composite Functions

Composite functions arise when one function's output becomes another function's input. Understanding them involves knowing both the basic functions and their allowable operations. A composite function is typically represented as $f(g(x))$, indicating that $g(x)$ is inserted into $f(x)$.
For instance:

$\cos(\ln x)$ is a composition, where $\ln x$ feeds into the cosine function.
$\sin^3(x)$ can be seen as composing $\sin(x)$ and raising the result to the third power.

The continuity of composite functions depends on the continuity of both the inner and outer functions over applicable domains. In this case, both are continuous, ensuring that their composite is as well.

Product of Functions

The product of functions involves multiplying two or more functions together to create a new one. This multiplication can create complex behavior, but retaining continuity is possible, provided the participating functions are individually continuous.
In the given exercise:

$\sin^3(x) \cdot \tan^{-1}(x)$ is the product of the cube of the sine function and the inverse tangent function.

Each function in this multiplication is continuous in $\mathbb{R}$, so their product is also continuous. This property is vital for assessing the whole function's continuity, ensuring no breaks or undefined points where inputs are valid.

Problem 38

Problem 40

Other exercises in this chapter

Problem 37

Show that a polynomial function $P_{n}(x)=a_{n} x^{n}+a_{s-1} x^{n-1}+\ldots \ldots+a_{1} x+a_{0}$ is a continuous function.

View solution

Problem 38

Show that a rational function is a continuous function.

View solution

Problem 40

Check the continuity of $|x|$ and $\operatorname{sgn} x$.

View solution

Problem 41

Test the following functions for continuity $$ \begin{aligned} \phi(x) &=0, \quad x=0 \\ &=\frac{1}{2}-x, \quad 0

View solution