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Chapter 2: Problem 17

Applying limit laws Assume \(\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3\) and \(\lim _{x \rightarrow 1} h(x)=2 .\) Compute the following limits and state the limit laws used to justify your computations. $$\lim _{x \rightarrow 1}[4 f(x)]$$

Short Answer

Expert verified
Question: Compute the limit of \(4f(x)\) as x approaches 1, given that \(\lim_{x\rightarrow 1} f(x) = 8\). Answer: The limit of \(4f(x)\) as \(x\) approaches 1 is 32.

Step by step solution

01

Use the given limit of f(x) as x approaches 1

We are given that \(\lim_{x\rightarrow 1} f(x) = 8\). Step 2: Use the limit constant multiple rule
02

Apply the limit constant multiple rule

The limit constant multiple rule states that \(\lim_{x\rightarrow a} k*f(x) = k * \lim_{x\rightarrow a} f(x)\) where k is a constant. Using this rule, we get: $$\lim _{x \rightarrow 1}[4 f(x)] = 4*\lim_{x\rightarrow 1} f(x)$$ Step 3: Substitute the known limit
03

Substitute the limit value of f(x)

Now, we substitute the given limit value of \(f(x)\) from Step 1: $$4*\lim_{x\rightarrow 1} f(x) = 4*8$$ Step 4: Compute the final result
04

Compute the limit value

Multiplying the constant by the limit value we get: $$4*8 = 32$$ So, the limit of \(4f(x)\) as \(x\) approaches 1 is 32: $$\lim _{x \rightarrow 1}[4 f(x)] = 32$$

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Key Concepts

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

Constant Multiple Rule
The Constant Multiple Rule is an essential tool in calculus that simplifies finding the limits of functions multiplied by a constant. When you have a function like \(f(x)\), and it's multiplied by a constant \(k\), you can isolate that constant outside the limit expression. This rule states that:
  • \( \lim_{x \to a} [k \cdot f(x)] = k \cdot \lim_{x \to a} f(x) \)
What does this mean? It means that if you know the limit of \(f(x)\) as \(x\) approaches \(a\), you can find the limit of \(k \cdot f(x)\) by simply multiplying that limit by \(k\).
For example, if \( \lim_{x \to 1} f(x) = 8 \) and you have an expression \(4f(x)\), then:
  • \( \lim_{x \to 1} [4f(x)] = 4 \cdot 8 = 32 \)
This makes complex problems manageable and helps us compute limits more efficiently.
Limit of a Function
Understanding the limit of a function is key in calculus. It tells us the value that a function approaches as the input approaches a particular point. Mathematically, it's written as \( \lim_{x \to a} f(x) \).To grasp this concept:
  • Think of a car approaching a stop sign. The limit helps us find the speed of the car as it gets closer to the sign, even if it never quite stops.
  • Your goal is to find the output (or limit) value as \(x\) gets infinitely close to a specific \(a\).
In the given exercise, we have known limits for functions \(f(x)\), \(g(x)\), and \(h(x)\) as \(x\) approaches 1:
  • \( \lim_{x \to 1} f(x) = 8 \)
  • \( \lim_{x \to 1} g(x) = 3 \)
  • \( \lim_{x \to 1} h(x) = 2 \)
These values become the basis for applying limit laws in various calculations.
Basic Calculus Concepts
Basic calculus concepts are foundational for understanding more complex mathematical ideas. They include limits, derivatives, and integrals. **Calculus Basics:**
  • Limits: As explained earlier, help determine the value a function approaches.
  • Derivatives: These measure how a function changes as its input changes. Think of it as the slope of a function at any given point.
  • Integrals: These are about finding areas under curves, which can be related to accumulated quantities.
In our exercise, we deal specifically with limits.
Over time, understanding these core concepts will allow you to solve complex calculus problems with ease.
Connecting each part, like using the Constant Multiple Rule efficiently, builds a strong foundation for future studies.

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Most popular questions from this chapter

Refer to Exercises \(91-92.\) a. Does the function \(f(x)=x \sin (1 / x)\) have a removable discontinuity at \(x=0 ?\) b. Does the function \(g(x)=\sin (1 / x)\) have a removable discontinuity at \(x=0 ?\)

Find polynomials \(p\) and \(q\) such that \(p / q\) is undefined at 1 and \(2,\) but \(p / q\) has a vertical asymptote only at \(2 .\) Sketch a graph of your function.

A monk set out from a monastery in the valley at dawn. He walked all day up a winding path, stopping for lunch and taking a nap along the way. At dusk, he arrived at a temple on the mountaintop. The next day, the monk made the return walk to the valley, leaving the temple at dawn, walking the same path for the entire day, and arriving at the monastery in the evening. Must there be one point along the path that the monk occupied at the same time of day on both the ascent and descent? (Hint: The question can be answered without the Intermediate Value Theorem.) (Source: Arthur Koestler, The Act of Creation.)

A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence \(\\{2,4,6,8, \ldots\\}\) is specified by the function \(f(n)=2 n\), where \(n=1,2,3, \ldots .\) The limit of such a sequence is \(\lim _{n \rightarrow \infty} f(n)\), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences, or state that the limit does not exist. \(\left\\{0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots .\right\\},\) which is defined by \(f(n)=\frac{n-1}{n},\) for \(n=1,2,3, \ldots\)

Use the following definitions. Assume fexists for all \(x\) near a with \(x>\) a. We say that the limit of \(f(x)\) as \(x\) approaches a from the right of a is \(L\) and write \(\lim _{x \rightarrow a^{+}} f(x)=L,\) if for any \(\varepsilon>0\) there exists \(\delta>0\) such that $$ |f(x)-L|<\varepsilon \quad \text { whenever } \quad 00\) there exists \(\delta>0\) such that $$ |f(x)-L|<\varepsilon \quad \text { whenever } \quad 0
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