/*! 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 21 Find the limit. $$ \lim _{x ... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 21

Find the limit. $$ \lim _{x \rightarrow-\infty} \frac{3 x+4}{2 x-3} $$

Short Answer

Expert verified
The limit of the function as x approaches negative infinity is \(\frac{3}{2}\).

Step by step solution

01

Identify the degree of the numerator and the denominator.

Here, we are given a rational function: \(f(x) = \frac{3x+4}{2x-3}\) Let's identify the degree of the numerator and the denominator: - The numerator (3x + 4) has a degree of 1, since the highest power of x in the numerator is x^1. - The denominator (2x - 3) also has a degree of 1, as the highest power of x in the denominator is x^1.
02

Understand how to deal with limits involving rational functions

For a rational function \(g(x) = \frac{P(x)}{Q(x)}\), where P(x) and Q(x) are polynomials, we can find the limit as x approaches positive or negative infinity by observing the degree of P(x) and Q(x): 1. If the degree of P(x) is less than the degree of Q(x), the limit is 0. 2. If the degree of P(x) is greater than the degree of Q(x), the limit is infinity or negative infinity, depending on the sign of the function as x approaches positive or negative infinity. 3. If the degree of P(x) is equal to the degree of Q(x), the limit is equal to the ratio of the leading coefficients. In our case, the degree of P(x) is equal to the degree of Q(x) (both degrees are 1).
03

Apply the appropriate technique to find the limit

Since the degree of the numerator is equal to the degree of the denominator, we will apply the third case by finding the ratio of the leading coefficients. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 2. Therefore, the limit of the function as x approaches negative infinity is: \(\lim_{x \rightarrow -\infty} \frac{3x+4}{2x-3} = \lim_{x \rightarrow -\infty} \frac{3}{2}\) Hence, the limit of the function as x approaches negative infinity is \(\frac{3}{2}\).

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Key Concepts

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

Limits at Infinity
When studying calculus, students often encounter the concept of limits at infinity, which deals with understanding the behavior of functions as the input approaches positive or negative infinity. Simplifying, it's like asking ourselves, what value does the function approach when the input grows larger and larger (or decreases without bound)?

An example of this is the equation from the original exercise: \(\lim _{x \rightarrow-\infty} \frac{3 x+4}{2 x-3} \)
This limit asks what happens to our function as x becomes very large in the negative direction. To determine this, we look at the degrees and leading coefficients of the polynomials in the numerator and the denominator.
Rational Function Behavior
Rational functions, which are ratios of polynomials, have certain behaviors at infinity that can be anticipated based on the degrees of these polynomials. The step-by-step solution explains that we're dealing with three distinct cases. In the exercise function \(\frac{3x+4}{2x-3} \), both the numerator and denominator have the same degree, meaning they both have the power of x to the first degree. Consequently, the behavior of this function at infinity can be understood just by comparing their leading coefficients. For a more visual grasp, students could plot the function using graphing software to see that, indeed, as x approaches negative infinity, the graph levels out toward a particular value.
Leading Coefficients
Leading coefficients are the front-runners in the race to infinity; they're the coefficients of the term with the highest power of x in a polynomial. In the example \(\frac{3x+4}{2x-3} \), the leading coefficient of the numerator is 3, and for the denominator, it is 2.

These numbers are crucial when the degrees of the numerator and denominator match, as they determine the horizontal asymptote of the function's graph. This is the value that the function will get closer and closer to but never quite reach as x heads towards infinity (positively or negatively). Here's a tip: when the degrees are equal, the limit at infinity becomes the ratio of these leading coefficients.
Degrees of Polynomials
The degree of a polynomial is the highest power of the variable x that appears in the polynomial. It's a significant concept when dealing with limits at infinity for rational functions. If you have a higher degree in the numerator than the denominator, the function will go off to infinity or negative infinity. If the denominator's degree is greater, the function gets closer to zero.

The moment their degrees are equal, just like in our exercise with the polynomials \(3x+4 \), and \(2x-3 \), the function will approach a horizontal line at the value of their leading coefficient's ratio as x grows large. Remember, higher degree means the term will grow faster and dominate the function's behavior as x approaches infinity.

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Most popular questions from this chapter

Use Newton's method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than \(0.0001\). The zero of \(f(x)=5 x+\cos x-5\) between \(x=0\) and \(x=1\). Take \(x_{0}=0.5\).

Use Newton's method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than \(0.0001\). The zero of \(f(x)=x^{5}+2 x^{4}+2 x-4\) between \(x=0\) and \(x=1\). Take \(x_{0}=0.5\).

Terminal Velocity A skydiver leaps from a helicopter hovering high above the ground. Her velocity \(t\) sec later and before deploying her parachute is given by $$ v(t)=52\left[1-(0.82)^{l}\right] $$ where \(v(t)\) is measured in meters per second. a. Complete the following table, giving her velocity at the indicated times. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline t \text { (sec) } & 0 & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline \boldsymbol{p}(t)(\mathrm{m} / \mathrm{sec}) & & & & & & & \\ \hline \end{array} $$ b. Plot the graph of \(v\) using the viewing window \([0,60] \times[0,60]\) c. What is her terminal velocity?

Determine whether the given statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If the denominator of a rational function \(f\) is equal to zero at \(a\), then \(x=a\) is a vertical asymptote of the graph of \(f\).

Resonance A spring system comprising a weight attached to a spring and a dashpot damping device (see the accompanying figure) is acted on by an oscillating external force. Its motion for large values of \(t\) is described by the equation $$ x(t)=\frac{F}{\left(\omega^{2}-\gamma\right)^{2}+4 \lambda^{2} \gamma^{2}} \sin (\gamma t+\theta) $$ where \(F, \omega, \lambda\), and \(\theta\) are constants. \((F\) is the amplitude of the external force, \(\theta\) is a phase angle, \(\gamma\) is associated with the frequency of the external force, and \(\omega\) and \(\lambda\) are associated with the stiffness of the spring and the degree of resistance of the dashpot damping device, respectively.) Show that the amplitude of the motion of the system $$ g(\gamma)=\frac{F}{\sqrt{\left(\omega^{2}-\gamma^{2}\right)^{2}+4 \lambda^{2} \gamma^{2}}} $$ has a maximum value at \(\gamma_{1}=\sqrt{\omega^{2}-2 \lambda^{2}}\). When the frequency of the external force is \(\sqrt{\omega^{2}-2 \lambda^{2}} / 2 \pi\), the system is said to be in resonance. The figure below shows a typical resonance curve. The external force imparts an oscillatory vertical motion on the support.
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