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Chapter 1: Problem 72

Describe the interval(s) on which the function is continuous. $$ f(x)=\frac{x+1}{\sqrt{x}} $$

Short Answer

Expert verified
The function is continuous on the intervals \(-\infty<x<0\) and \(0<x<\infty\).

Step by step solution

01

Define the function

The function is defined as \(f(x)=\frac{x+1}{\sqrt{x}}\). We need to determine the interval on which this function is continuous.
02

Identify Discontinuities

Discontinuities in a function typically occur where the denominator equals zero, because division by zero is undefined. In this function, the denominator is \(\sqrt{x}\), which equals zero when \(x=0\). This means that the function is discontinuous at \(x=0\).
03

Determine continuity interval

The function will be continuous everywhere else except at the points of discontinuity. The resulting intervals of continuity are \(-\infty<x<0\) and \(0<x<\infty\) which means the function is continuous for all real numbers except 0.

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Key Concepts

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

Discontinuities
When we talk about discontinuities in calculus, we are referring to points at which a function is not continuous. These discontinuities can occur for a variety of reasons, but one common reason is when a function involves a fraction with a denominator that can become zero. In such instances, division by zero is not defined, leading to a 'gap' in the function.

Using our example, the function \(f(x)=\frac{x+1}{\sqrt{x}}\) has a discontinuity where the denominator, \(\sqrt{x}\), is equal to zero which occurs when \(x=0\). Thus, at \(x=0\), the function cannot produce a value and is said to be discontinuous. Discontinuities are critical to understand because they affect the domain of the function and its potential for integration and differentiation.
Interval of Continuity
The interval of continuity refers to the range of values over which a function is continuous. In between points of discontinuity, a function can operate without interruption. This is important for understanding the behaviour of the function across its domain.

In our step-by-step solution, the interval of continuity for the function \(f(x)=\frac{x+1}{\sqrt{x}}\) is \(-\infty
Functions in Calculus
In calculus, functions are mathematical expressions that consistently associate elements of one set with elements of another. A function can be represented by equations, graphs, or formulas and is fundamental for calculus operations such as differentiation and integration.

The function \(f(x)=\frac{x+1}{\sqrt{x}}\) is an example of a rational function, which is a ratio of two polynomials. Rational functions are one type of function you'll encounter in calculus. Understanding the behavior and characteristics of functions, including their continuity and discontinuity, is crucial in calculus as it affects the computation of limits, derivatives, and integrals - all of which are foundational concepts in the study of calculus.

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Most popular questions from this chapter

Discuss the continuity of each function. $$ f(x)=\frac{x^{2}-1}{x+1} $$

Consider \(f(x)=\frac{\sec x-1}{x^{2}}\). (a) Find the domain of \(f\). (b) Use a graphing utility to graph \(f .\) Is the domain of \(f\) obvious from the graph? If not, explain. (c) Use the graph of \(f\) to approximate \(\lim _{x \rightarrow 0} f(x)\). (d) Confirm the answer in part (c) analytically.

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Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 4^{-}} \frac{\sqrt{x}-2}{x-4} $$
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