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Chapter 3: Problem 17

Finding Limits at Infinity In Exercises \(15-18\) , find each limit, if possible. $$ \begin{array}{l}{\text { (a) } \lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x^{2}-4}} \\ {\text { (b) } \lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x^{3 / 2}-4}} \\ {\text { (c) } \lim _{x \rightarrow \infty} \frac{5-2 x^{3 / 2}}{3 x-4}}\end{array} $$

Short Answer

Expert verified
The limits for (a), (b), and (c) are all 0

Step by step solution

01

Solve (a)

Using the concept of limits, which states that \( \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow \infty} f(x)}{\lim_{x \rightarrow \infty} g(x)} \) if both limits in the right hand exist and the denominator is not equal to zero. We apply this expression to calculate the first limit. The top limit is \( \lim_{x \rightarrow \infty} (5-2x^{3/2}) \), while the bottom limit is \( \lim_{x \rightarrow \infty} (3x^2 - 4) \). Both limits approach negative infinity, so applying L'hopital rule as the forms mimic \( \frac{\infty}{\infty} \). Taking derivatives of both the top and bottom until a determinable limit can be found, we find that the solution is 0. Apply same process to remainder of exercises.
02

Solve (b)

Using the same concept as in Step 1, we calculate the limit of the second expression. The top limit is \( \lim_{x \rightarrow \infty} (5-2x^{3/2}) \), while the bottom limit is \( \lim_{x \rightarrow \infty} (3x^{3/2} - 4) \). Applying L'hopital rule, we find that the solution is also 0.
03

Solve (c)

Finally, for the third expression, the top limit is \( \lim_{x \rightarrow \infty} (5-2x^{3/2}) \), while the bottom limit is \( \lim_{x \rightarrow \infty} (3x - 4) \). Lhopital's rule here also reveals that the limit of the ratio is 0.

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Key Concepts

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

limits
Understanding limits is fundamental in calculus. They describe the value that a function approaches as the input (or index) approaches some value. Limits can be at a specific point, such as when a function approaches a particular 'x' value, or at infinity, which tells us the behavior of a function as 'x' becomes very large or very small (negatively large).

When evaluating limits, it's crucial to recognize different forms, like when the result of a limit appears to be a fraction where the numerator and denominator both approach infinity, known as an indeterminate form. In such cases, we sometimes need additional tools like L'Hopital's rule to properly evaluate the limit.
L'Hopital's rule

Applying L'Hopital's Rule

L'Hopital's rule is a technique we use to find limits of indeterminate forms like \(0/0\) or \(\infty/\infty\). It says that if the limits of the functions in the numerator \(f(x)\) and the denominator \(g(x)\) both approach 0 or both approach \(\infty\) as 'x' approaches a particular value, then the original limit is equal to the limit of their derivatives ratio:

\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]

If this new limit is still indeterminate, we can apply L'Hopital's rule repeatedly until we obtain a determinate form or show that the limit does not exist.
infinite limits calculus
In calculus, infinite limits refer to the behavior of functions as the independent variable approaches infinity or negative infinity. These limits help us understand the long-term behaviour of functions, including growth rates and end-behavior modeling.

For example, in the given exercise, we investigate the limits as 'x' tends towards infinity. As 'x' becomes very large, the terms that grow fastest dominate the function's behavior. In expressions of the form \(x^n\), higher powers of 'x' outweigh lower powers, constants become negligible, and we're often left evaluating the ratio of the leading terms, simplifying the limit's calculation.
calculus exercises

Practical Application of Calculus Exercises

Working through calculus exercises, like finding limits at infinity, allows students to grasp the theoretical concepts and apply them to practical problems. The exercises demonstrate how calculus can describe real-world situations such as optimization, motion, and growth.

Completing exercises improves understanding and proficiency in using important calculus tools like L'Hopital's rule. It's essential for students to not just memorize the steps but to understand why each step is taken. This ensures they can tackle new and possibly more complex problems beyond those found in the textbook.

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Most popular questions from this chapter

Area The measurement of the side of a square floor tile is 10 inches, with a possible error of \(\frac{1}{32}\) inch. (a) Use differentials to approximate the possible propagated error in computing the area of the square. (b) Approximate the percent error in computing the area of the square.

Find the minimum value of $$\frac{(x+1 / x)^{6}-\left(x^{6}+1 / x^{6}\right)-2}{(x+1 / x)^{3}+\left(x^{3}+1 / x^{3}\right)}\( for \)x>0$$

Analyzing a Graph Using Technology In Exercises \(75-82,\) use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{1}{x^{2}-x-2} $$

Volume and Surface Area The radius of a spherical balloon is measured as 8 inches, with a possible error of 0.02 inch. (a) Use differentials to approximate the possible propagated (a) Use differentials the volume of the sphere. (b) Use differentials to approximate the possible propagated error in computing the surface area of the sphere. (c) Approximate the percent errors in parts (a) and (b).

Numerical, Graphical, and Analytic Analysis Find two positive numbers whose sum is 110 and whose product is a maximum. (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) $$ \begin{array}{|c|c|c|}\hline \text { First } & {\text { Second }} \\ {\text { Number, } x} & {\text { Number }} & {\text { Product, } P} \\ \hline 10 & {110-10} & {10(110-10)=1000} \\ \hline 20 & {110-20} & {20(110-20)=1800} \\\ \hline\end{array} $$ (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the solution. (Hint: Use the table feature of the graphing utility.) (c) Write the product \(P\) as a function of \(x\) . (d) Use a graphing utility to graph the function in part (c) and estimate the solution from the graph. (e) Use calculus to find the critical number of the function in part (c). Then find the two numbers.
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