/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Find the limit. $$ \lim _{x ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 6

Find the limit. $$ \lim _{x \rightarrow 7} 100 $$

Short Answer

Expert verified
The limit is 100.

Step by step solution

01

Identifying the Function and Limit

The given function is a constant function, defined as \( f(x) = 100 \). We need to find the limit of this function as \( x \) approaches 7.
02

Understanding Constant Functions

A constant function is a function that takes the same value for any input \( x \). Therefore, it does not change as \( x \) approaches any value, including 7.
03

Applying the Limit Rule for Constants

The limit of a constant function \( f(x) = c \) as \( x \) approaches any number \( a \) is \( c \). Thus, \( \lim_{x \to 7} 100 = 100 \).

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Key Concepts

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

Constant Function
A constant function is quite unique because, unlike most functions, it always gives the same output, no matter what input you provide. Imagine a machine that always snaps a perfectly identical photo鈥攏o matter who's in front of the camera, like a photo booth that always produces a picture of an apple. No twists or surprises!

Mathematically, a constant function is written as:
  • \( f(x) = c \)
  • Where "\(c\)" is the constant value, and it doesn鈥檛 change regardless of what "\(x\)" is.
In the given example, the function is \( f(x) = 100 \). This implies that for any value of \(x\), the output remains 100. It's a horizontal line on a graph, hovering at the value of the constant.
Limit Rules
When studying calculus, limit rules become vital tools, allowing us to investigate how a function behaves as it approaches a particular value. Knowing these rules can simplify complex problems and make them much easier to tackle.

In the case of a constant function, the limit rule is refreshingly simple.
  • **Limit of a Constant Rule**: If \( f(x) = c \) is a constant function and \( x \) approaches any value \( a \), then:
  • \[ \lim_{x \to a} c = c \]
This rule means that no matter what value \(x\) is approaching, the limit of a constant function is simply the constant itself. It's as straightforward as it sounds!
Approaching a Value
The concept of "approaching a value" in calculus is a fascinating journey of a function as it gets closer to a specific number or point. Think of it like slowly walking towards a door鈥攕tep by step, getting closer, but not necessarily crossing the threshold.

In calculus, we are interested in what happens as "\(x\)" gets infinitely close to a particular point, denoted by \( a \). In our scenario, we look at what happens when \(x\) approaches 7.
  • Even though \(x\) never reaches exactly 7, we can predict the behavior of the function as if it does.
  • For the constant function \(f(x) = 100\), this means staying at 100, regardless of the approach.
This exploration helps us understand functions better, especially in cases where the function might behave unusually just before reaching the value.

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Most popular questions from this chapter

Use theorems on limits to find the limit, if it exists. $$ \lim _{x \rightarrow 5^{+}} \frac{1+\sqrt{2 x-10}}{x+3} $$

Sketch the graph of \(f\) and find each limit, if it exists: (a) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 1^{+}} f(x)\) (c) \(\lim _{x \rightarrow 1} f(x)\) $$ f(x)=\left\\{\begin{array}{ll} x^{3} & \text { if } x \leq 1 \\ 3-x & \text { if } x>1 \end{array}\right. $$

Sketch the graph of \(f\) and find each limit, if it exists: (a) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 1^{+}} f(x)\) (c) \(\lim _{x \rightarrow 1} f(x)\) $$ f(x)=\left\\{\begin{array}{ll} |x-1| & \text { if } x \neq 1 \\ 1 & \text { if } x=1 \end{array}\right. $$

Let []\(]\) denote the greatest integer function and \(n\) an arbitrary integer. Find (a) \(\lim _{x \rightarrow n^{-}} f(x)\) (b) \(\lim _{x \rightarrow n^{+}} f(x)\) $$ f(x)=-[-x] $$

Find each limit, if it exists: (a) \(\lim _{x \rightarrow a^{-}} f(x)\) (b) \(\lim _{x \rightarrow a^{+}} f(x)\) (c) \(\lim _{x \rightarrow a} f(x)\) $$ f(x)=\sqrt{5-x} ; \quad a=5 $$
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