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Chapter 1: Problem 48

Find each of the right-hand and left-hand limits or state that they do not exist. $$ \lim _{x \rightarrow 3^{+}}\left[x^{2}+2 x\right] $$

Short Answer

Expert verified
The right-hand limit as \( x \to 3^+ \) is 15.

Step by step solution

01

Understanding the One-Sided Limit

The given limit expression is \( \lim_{x \to 3^+} [x^2 + 2x] \). This denotes the right-hand limit as \( x \) approaches 3 from values greater than 3. We need to substitute values into the expression \( x^2 + 2x \) from the right side of 3.
02

Substitute x with Values Close to 3 from the Right

To find \( \lim_{x \to 3^+} [x^2 + 2x] \), consider values like \( x = 3.01, 3.001, \text{and} \ 3.0001 \). These are slightly greater than 3.
03

Calculate the Expression for Each Value

For \( x = 3.01 \), calculate \( (3.01)^2 + 2(3.01) = 9.0601 + 6.02 = 15.0801 \). For \( x = 3.001 \), it is \( (3.001)^2 + 2(3.001) = 9.006001 + 6.002 = 15.008001 \). For \( x = 3.0001 \), it is \( (3.0001)^2 + 2(3.0001) = 9.00060001 + 6.0002 = 15.00080001 \).
04

Observe the Pattern

As the values of \( x \) approach 3 from the right, the computed expression \( x^2 + 2x \) approaches 15.
05

Conclude the Limit

Based on the pattern and calculations, we conclude that \( \lim_{x \to 3^+} [x^2 + 2x] = 15 \).

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Key Concepts

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

Right-Hand Limit
In calculus, when evaluating the limit of a function as the variable approaches a specific value from the right side, we refer to this as a right-hand limit. This is denoted by the notation \( \lim_{x \to a^+} \), where \( a^+ \) indicates that \( x \) is approaching \( a \) from values greater than \( a \). The right-hand limit of \( f(x) \) at a point evaluates how the function behaves as \( x \) comes from the right.

Understanding the right-hand limit can be important for discovering how functions behave at boundaries and understanding potential discontinuities. In some cases, the right-hand limit can exist even if the overall two-sided limit does not.
One-Sided Limits
One-sided limits refer to the evaluation of limits from only one direction鈥攅ither from the left or the right. A right-hand limit is a specific type of one-sided limit, denoted \( \lim_{x \to a^+} \), while the left-hand limit is denoted \( \lim_{x \to a^-} \).

When calculating one-sided limits:
  • We only consider numbers arbitrarily close to \( a \) from one side.
  • This concept helps determine the function's behavior at a certain point.
  • If both one-sided limits exist and are equal, the two-sided limit exists.
  • If they differ, the two-sided limit at that point does not exist.
One-sided limits provide a more complete picture of how various functions interact with points on a graph. These are crucial for analyzing points of discontinuity.
Substituting Values
Substituting values near the point of interest is a crucial step in evaluating a limit, especially for right-hand or left-hand limits. This process involves choosing numbers that grow closer and closer to the target point from the specified direction.

For example:
  • If evaluating \( \lim_{x \to 3^+} \), choose values like 3.01, 3.001, 3.0001.
  • Calculate the expression using these values to observe the trend as \( x \) gets closer to the number.
  • Observe if the function approaches a specific value consistently as it nears the limit point.
These calculations help in seeing the pattern and confirming the hypothesis about the limit. By substituting these values, one can gain a clearer understanding of the function's behavior near the point of interest, leading to a more accurate conclusion of the limit.

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Most popular questions from this chapter

In Problems \(18-23,\) the given function is not defined at a certain point. How should it be defined in order to make it continuous at that point? (See Example 1.) $$ f(x)=\frac{x^{2}-49}{x-7} $$

Prove that if \(f(x)\) is a continuous function on an interval then so is the function \(|f(x)|=\sqrt{(f(x))^{2}}\)

A stretched elastic string covers the interval [0,1] . The ends are released and the string contracts so that it covers the interval \([a, b], a \geq 0, b \leq 1\). Prove that this results in at least one point of the string being where it was originally. See Problem \(59 .\)

In Problems \(1-15,\) state whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ h(x)=\frac{3}{x-3} $$

$$ \text { Prove using an } \varepsilon-\delta \text { argument that } \lim _{x \rightarrow 3}(2 x+1)=7 \text { . } $$
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