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Chapter 11: Problem 7

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. $$\lim _{x \rightarrow-2} 7 x^{2}$$

Short Answer

Expert verified
The limit as x approaches -2 of the function \(7x^2\) is 28.

Step by step solution

01

Identify the Function and the Value of x

The function that is given is \(7x^2\), and the value of x that we are given is -2.
02

Substitute the Value of x into the Function

To find the limit as x approaches -2, substitute -2 into the function to get \(7(-2)^2\).
03

Simplify the Expression

Solving \(7(-2)^2\) gives you \(7 \times 4\), which simplifies to 28.

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Key Concepts

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

Limit Properties
When dealing with limits, it's crucial to understand the properties that simplify the process. Limits help us find the value that a function approaches as the input approaches a certain point. Several properties make solving these problems easier:
  • Sum and Difference Limitation: The limit of a sum or difference equals the sum or difference of the limits.
  • Constant Factor: The limit of a constant multiplied by a function is the constant multiplied by the limit of the function.
  • Power and Root Rule: The limit of a function raised to a power is the limit of the function raised to that power.
In our example, we utilize the constant factor property because the function is a constant multiple of a simpler expression. That enables us to separately consider the limit of the term inside the function and then multiply by the constant. This makes the calculation straightforward.
Substitution
Substitution is a handy technique when finding limits. It allows us to directly plug in the value to which the variable approaches into the expression:
  • This method works seamlessly if the function is continuous around that point.
  • Ensure that substituting doesn't lead to undefined operations, like division by zero.
In our example, the function given is a polynomial, which is continuous everywhere. This continuity means we can freely substitute the value of \(x = -2\) directly into \(7x^2\). So, we calculate \(-2 \) squared and multiply by 7. This direct approach is simple and effective, especially when handling polynomials.
Simplifying Expressions
Simplifying expressions is often the final step in evaluating limits. This process involves performing arithmetic operations and applying algebraic rules to make expressions more manageable. Let's break it down:
  • Handle Exponents: Start by calculating powers or any exponentiated terms.
  • Multiply Terms: Once the power is computed, perform any multiplications.
  • Simplify: Finally, bring everything together for a clear and concise result.
In this problem, the expression becomes straightforward after substitution. We calculate \(7 imes (-2)^2 = 7 imes 4\), which ultimately simplifies to 28. This calculated result represents the limit as \(x\) approaches \(-2\). Simplifying right after substitution ensures clarity and a correct final value for the limit.

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Most popular questions from this chapter

Consider the function \(f(x)=3 x+2 .\) As \(x\) approaches 1 \(f(x)\) approaches \(5: \lim _{x \rightarrow 1} f(x)=5 .\) Find the values of \(x\) such that \(f(x)\) is within 0.1 of 5 by solving $$ |f(x)-5|<0.1 $$ Then find the values of \(x\) such that \(f(x)\) is within 0.01 of 5

In this exercise, the group will define three piecewise functions. Each function should have three pieces and two values of \(x\) at which the pieces change. a. Define and graph a piecewise function that is continuous at both values of \(x\) where the pieces change. b. Define and graph a piecewise function that is continuous at one value of \(x\) where the pieces change and discontinuous at the other value of \(x\) where the pieces change. c. Define and graph a piecewise function that is discontinuous at both values of \(x\) where the pieces change. At the end of the activity, group members should turn in the functions and their graphs. Do not use any of the piecewise functions or graphs that appear anywhere in this book.

Graph each function. Then determine for what numbers, if any, the function is discontinuous. $$f(x)=\left\\{\begin{aligned}2 & \text { if } x \text { is an odd integer. } \\\\-2 & \text { if } x \text { is not an odd integer. }\end{aligned}\right.$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. \(f\) and \(g\) are both continuous at \(a\), although \(f+g\) is not.

Will help you prepare for the material covered in the next section. In each exercise, use what occurs near 3 and at 3 to draw the graph of a function \(f\) (Graphs will vary.) Is it necessary to lift your pencil off the paper to obtain graph? Explain your answer. $$\lim _{x \rightarrow 3^{-}} f(x)=5 ; \lim _{x \rightarrow 3^{+}} f(x)=6 ; f(3)=5$$
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