/*! 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 48 In Exercises 47 and 48, find $... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 48

In Exercises 47 and 48, find $$\textrm{(a)}\ \lim_{x \to 2}\ f(x), \quad \textrm{(b)} \lim_{x \to 2}\ g(x), \quad \textrm{(c)} \lim_{x \to 2}\ [f(x)g(x)], \quad \textrm{and (d)} \lim_{x \to 2}\ [g(x)-f(x)].$$ $$f(x)=\dfrac{x}{3-x}, \quad \quad g(x)=\sin \pi x$$

Short Answer

Expert verified
Therefore, \(\lim_{x \to 2}\ f(x) = 2\), \(\lim_{x \to 2}\ g(x) = 0\), \(\lim_{x \to 2}\ [f(x)g(x)] = 0\) and \(\lim_{x \to 2}\ [g(x)-f(x)] = -2\)

Step by step solution

01

Find the limit of f(x) as x approaches 2

Substitute x with 2 in the function f(x) = x/(3-x), \n Resulting in f(2) = 2/(3-2) which is equal to 2.
02

Find the limit of g(x) as x approaches 2

Substitute x with 2 in the function g(x) = sin(\(\pi x\)), \n Resulting in g(2) = sin(\(\pi * 2\)) which is equal to 0 as sin(\(2 \pi \)) is always equals to 0.
03

Find the limit of f(x)g(x) as x approaches 2

Multiplication of the two functions f(x) and g(x) at x=2, \n Resulting in f(2)g(2) = 2*0 which is equal to 0.
04

Find the limit of g(x)-f(x) as x approaches 2

Subtraction of the two functions f(x) and g(x) at x=2, \n Resulting in g(2)-f(2) = 0-2 which is equal to -2.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Function Operations
Function operations involve combining functions using arithmetic operations such as addition, subtraction, multiplication, and division. This exercise asks you to compute limits using these operations. When calculating limits, remember you're finding the behavior of a function as it approaches a certain point without necessarily reaching that point.
  • For instance, in this exercise, examining the limit of \((f(x)g(x))\) involves multiplying individual limits. If these limits exist, \((\lim_{{x \to 2}} f(x)g(x) = \lim_{{x \to 2}} f(x) \cdot \lim_{{x \to 2}} g(x))\), provided neither is infinite or undefined.
  • Similarly, the limit of a difference, such as \((g(x) - f(x))\), corresponds to subtracting limits: \((\lim_{{x \to 2}} g(x) - \lim_{{x \to 2}} f(x))\).
Understanding these operations is crucial to evaluating limits of composite expressions.
Trigonometric Functions
Trigonometric functions are fundamental to understanding waves, circles, and oscillations, and they appear frequently in calculus problems. In this exercise, we have a trigonometric function \((g(x) = \sin(\pi x))\).
  • The sine function, \(\sin(\theta)\), represents the y-coordinate of a point on the unit circle for an angle \(\theta\). It oscillates between -1 and 1.
  • At specific important angles, like \(0, \pi, 2\pi\), the sine function equals 0. This is fundamental when calculating the limit, as \(\sin(2\pi) = 0\).
Recognizing these properties allows you to simplify calculations and immediately understand behaviors of trigonometric functions in limit problems.
Evaluating Limits
Evaluating limits requires carefully analyzing how functions behave as input values approach a point. This can be done via substitution, factoring, or recognizing special function behavior.
  • Substitution is the most straightforward method, used when the function is defined and well-behaved at the target point. If direct substitution leads to an undefined form like \(\frac{0}{0}\), other techniques may be necessary.
  • In this exercise, substitution worked effectively, allowing us to compute \(\lim_{{x \to 2}} f(x)\) and \(\lim_{{x \to 2}} g(x)\).
Mastery of limits is foundational, as it serves as a stepping stone to more advanced calculus topics like differentiation and integration.
Rational Functions
Rational functions are fractions involving polynomials, such as \(f(x) = \frac{x}{3-x}\). They are characterized by the presence of a variable in the denominator.
  • When evaluating limits, always check where the denominator can become zero, as this indicates potential vertical asymptotes or undefined points.
  • With \(f(x)\), substitution yields \(2/(3-2) = 2\), since the function is not approaching a problematic undefined form at \(x = 2\).
Understanding the behavior of these functions enhances your ability to predict outcomes in limits and identify asymptotic behavior or potential discontinuities.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises 29-36, complete the table using the function \( f(x) \), over the specified interval [a, b], to approximate the area of the region bounded by the graph of the \( y = f(x) \), the x-axis, and the vertical lines \(x=a\) and \(x=b\) and using the indicated number of rectangles. Then find the exact area as \( n \to \infty \). $$ f(x) = 2x + 5 $$ Interval \( [0, 4] \)

OXYGEN LEVEL Suppose that \(f(t)\) measures the level of oxygen in a pond, where \(f(t) = 1\) is the normal (unpolluted) level and the time \(t\) is measured in weeks. When \(t=0\), organic waste is dumped into the pond, and as the waste material oxidizes, the level of oxygen in the pond is given by $$ f(t) = \dfrac{t^2 - t + 1}{t^2 + 1} $$. (a) What is the limit of \(f\) as \(t\) approaches infinity? (b) Use a graphing utility to graph the function and verify the result of part (a). (c) Explain the meaning of the limit in the context of the problem.

In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{i=1}^{60} 7$$

In Exercises 39-48, write the first five terms of the sequence and find the limit of the sequence (if it exists). If the limit does not exist, explain why. Assume \(n\) begins with 1. $$ a_n = \dfrac{4n-1}{n+3} $$

In Exercises 63-70, use the function and its derivative to determine any points on the graph of \(f\) at which the tangent line is horizontal. Use a graphing utility to verify your results. \(f(x) = x^2 e^x, \quad f'(x) = x^2 e^x + 2xe^x\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.