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Chapter 1: Problem 1

Classify each statement as either true or false. $$\lim _{x \rightarrow 3} 7=7$$

Short Answer

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True

Step by step solution

01

Understand the limit concept

A limit describes the value that a function approaches as the input approaches some value. In this case, we are examining the constant function where the output is 7 regardless of the input.
02

Evaluate the limit

For the constant function, the value of the function does not change regardless of the input. Therefore, as the input approaches any value, including 3, the function's output remains 7.
03

Restate the result

Since the value of the constant function is always 7, it is clear that \(\text{lim}_{x \rightarrow 3} 7 = 7\).
04

Classify the statement

Given the evaluation, it is clear that the statement provided is true.

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Key Concepts

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

Limit of a Constant Function
In calculus, a constant function is a function that always returns the same value, no matter what the input is. For example, the function \(f(x) = 7\) returns 7 for any value of \(x\). When you take the limit of a constant function as \(x\) approaches any value, the result is simply the constant itself.

This can be expressed mathematically as follows:
\[ \lim_{x \rightarrow c} k = k \]
Here:
  • \(k\) is the constant function.
  • \(c\) is the value that \(x\) is approaching.
So, \[ \lim_{x \rightarrow 3} 7 = 7 \]
is always true as the constant 7 does not change no matter what \(x\) approaches.
Evaluation of Limits
The process of evaluating limits involves finding the value that a function approaches as \(x\) gets closer to a specific point. There are different ways of evaluating limits:
  • Direct Substitution: Plug the value of \(x\) directly into the function.
  • Factoring: Simplify the function by factoring and canceling common terms.
  • L'H么pital's Rule: Used for indeterminate forms like \(\frac{0}{0}\) or \(\frac{\text{鈭瀩}{\text{鈭瀩}\).

In the case of a constant function like \(7\), evaluating the limit is straightforward because the function does not change with \(x\). Hence, the limit is simply the constant value.
Calculus Basics
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It has two main branches:
  • Differential Calculus: Deals with the concept of a derivative, which represents the rate of change.
  • Integral Calculus: Focuses on the accumulation of quantities and the areas under and between curves.

Understanding limits is a fundamental concept in calculus. It forms the basis for derivatives and integrals:
  • Limits help us understand the behavior of functions as they approach specific points.
  • They are essential for defining the exact slope of a function (derivative).
  • Limits are also used to calculate areas under curves (integrals).

By mastering limits, students build a strong foundation for further studies in calculus.

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Most popular questions from this chapter

In Exercise : a) Graph the function. b) Draw tangent lines to the graph at points whose \(x\) -coordinates are \(-2,0,\) and 1. c) Find \(f^{\prime}(x)\) by determining \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}.\) d) Find \(f^{\prime}(-2), f^{\prime}(0),\) and \(f^{\prime}(1) .\) These slopes should match those of the lines you drew in part ( \(b\) ). $$f(x)=x^{2}-x$$

The Candy Factory sells candy by the pound, charging \(\$ 1.50\) per pound for quantities up to and including 20 pounds. Above 20 pounds, the Candy Factory charges \(\$ 1.25\) per pound for the entire quantity, plus a quantity surcharge \(k .\) If \(x\) represents the number of pounds, the price function is \(p(x)=\left\\{\begin{array}{ll}1.50 x, & \text { for } x \leq 20 \\ 1.25 x+k, & \text { for } x>20\end{array}\right.\) a) Find \(k\) such that the price function \(p\) is continuous at \(x=20.\) b) Explain why it is preferable to have continuity at \(x=20.\)

The population of a city grows from an initial size of 100,000 to a size \(P\) given by \(P(t)=100,000+2000 t^{2}\) where \(t\) is in years. a) Find the growth rate, \(d P / d t\) b) Find the population after 10 yr. c) Find the growth rate at \(t=10\) d) Explain the meaning of your answer to part (c).

Differentiate each function. $$f(x)=\left(\frac{2 x}{x^{2}+1}\right)^{3}$$

Differentiate each function. $$y=(7-x)^{55}$$
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