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Chapter 5: Problem 20

Is the function defined by \(f(x)=x^{2}-\sin x+5\) continuous at \(x=\pi\) ?

Short Answer

Expert verified
The function \(f(x)=x^{2}-\sin(x)+5\) is continuous at \(x=\pi\).

Step by step solution

01

Understanding Continuity at a Point

A function is continuous at a point if the limit of the function as it approaches the point equals the function's value at that point. Formally, a function f(x) is continuous at x = c if 1. f(c) is defined, 2. The limit as x approaches c of f(x) exists, 3. The limit of f(x) as x approaches c is equal to f(c).
02

Evaluate the Function at the Point

Calculate the value of the function at x = \(\pi\). Plug \(\pi\) into the function: \[f(\pi)=\pi^{2}-\sin(\pi)+5\].We know that \(\sin(\pi)=0\), hence \[f(\pi)=\pi^{2}+5\].
03

Checking the Limit of the Function as x Approaches the Point

To prove continuity, we must show that the limit exists as x approaches \(\pi\). The function \(f(x) = x^{2} - \sin(x) + 5\) consists of two polynomials (\(x^{2}\) and 5) and the sine function. Both polynomials and the sine function are continuous everywhere on the real number line. Therefore, the sum of these functions is also continuous at every point, including at \(x = \pi\).
04

Conclude the Continuity

Since the function satisfies the three conditions for continuity at \(x = \pi\): f(\pi) is defined, the limit as x approaches \(\pi\) exists and is equal to the value of the function at that point. Thus, \(f(x)\) is continuous at \(x = \pi\).

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Key Concepts

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

Limits and Continuity
When discussing mathematics, particularly calculus, the concepts of limits and continuity are foundational and closely linked. A limit is essentially the value that a function, denoted as f(x),

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