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Chapter 3: Problem 8

The notation \(x \rightarrow 5^{-}\) is read as _____ .

Short Answer

Expert verified
x approaches 5 from the left side.

Step by step solution

01

- Understand the Notation

The notation \(x \rightarrow 5^{-}\) involves the variable x approaching the number 5 from the left. This means that x is getting closer and closer to 5, but always remains less than 5.
02

- Interpret the Symbol -

The symbol \(- \) in \(5^{-}\) indicates that x is approaching 5 from the side that is less than 5. This can also be referred to as approaching 5 from the negative direction or from the left.
03

- Combine the Interpretation

Putting the two parts together, \(x \rightarrow 5^{-}\) means that x is approaching the value 5 from values that are less than 5.

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Key Concepts

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

left-hand limit
In calculus, understanding the left-hand limit is fundamental. When we talk about the left-hand limit of a function as it approaches a certain value, we mean we're looking at the values of the function as the variable gets closer to that value from the left side. For instance, if we say \(x \rightarrow 5^{-}\), this means x is getting closer to 5 but is always slightly less than 5. Imagine walking towards a door from the left side, but you never actually reach the exact spot of the door; you're always just a tiny bit to the left.
approaching a value
In mathematics and particularly in calculus, approaching a value is a critical concept. It means getting closer and closer to a particular number, but not necessarily reaching it. For example, consider the notation \(x \rightarrow 5^{-}\). This means that x is getting closer to 5 from values less than 5. It's like stepping towards 5 on a number line but never quite reaching it. This concept helps in understanding how functions behave near specific points, which is crucial for studying limits and continuity.
notation in calculus
Notation in calculus is a way of expressing mathematical concepts concisely. For example, the expression \(x \rightarrow 5^{-}\) tells us that x is approaching 5 from the left side. The notation uses the arrow \( \rightarrow \) to signify approaching, and the minus sign \( - \) indicates from the left. Proper notation is essential because it allows students and mathematicians to communicate ideas clearly and unambiguously. Always pay attention to details like arrows and signs in calculus notation, as they provide significant information about the behavior of functions and variables.
calculus fundamentals
Understanding the fundamentals of calculus is key to mastering advanced concepts. Calculus studies how things change and is essential in fields like physics, engineering, and economics. One of the core ideas is the limit, which describes how a function behaves as its input gets closer to a certain point. For instance, learning what \(x \rightarrow 5^{-}\) means is all about grasping the limit concept. In this specific example, it tells us about behavior from the left side. These basics pave the way for more complex topics like derivatives and integrals, which all build on the idea of limits.

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Most popular questions from this chapter

Write an equation of a function that meets the given conditions. Answers may vary. \(x\) -intercept: \(\left(\frac{4}{3}, 0\right)\) vertical asymptotes: \(x=-3\) and \(x=-4\) horizontal asymptote: \(y=0\) \(y\) -intercept: (0,-1)

Suppose that \(y\) varies jointly as \(x^{4}\) and \(w\). If \(x\) is replaced by \(\frac{1}{4} x\) and \(w\) is replaced by \(4 w,\) what is the effect on \(y ?\)

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ \sqrt{3 x-5}-4<0 $$

Suppose that \(y\) varies directly as \(x^{2}\) and inversely as \(w^{4}\). If both \(x\) and \(w\) are doubled, what is the effect on \(y\) ?

Use the rational zero theorem to show that \(\sqrt{5}\) is an irrational number. (Hint: Show that \(f(x)=x^{2}-5\) has no rational zeros.)
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