Textbook Question
Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x)=3e^x+10 / e^x
Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 2, Problem 79
A right circular cylinder with a height of 10 cm and a surface area of S cm2 has a radius given by r(S)=1/2(√100+2S/π −10).
Find lim S→0^+ r(S) and interpret your result.
Verified step by step guidance1
Identify the function for the radius: \( r(S) = \frac{1}{2}(\sqrt{100 + \frac{2S}{\pi}} - 10) \).
Understand that you need to find \( \lim_{S \to 0^+} r(S) \).
Substitute \( S = 0 \) into the function: \( r(0) = \frac{1}{2}(\sqrt{100 + \frac{2 \cdot 0}{\pi}} - 10) \).
Simplify the expression: \( r(0) = \frac{1}{2}(\sqrt{100} - 10) \).
Calculate the limit by evaluating the simplified expression.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
In calculus, a limit describes the behavior of a function as its input approaches a certain value. It is essential for understanding continuity, derivatives, and integrals. In this context, we are interested in the limit of the radius function r(S) as S approaches 0 from the positive side, which helps us determine the behavior of the radius when the surface area is minimal.
Recommended video:
05:50
One-Sided Limits
Surface Area of a Cylinder
The surface area of a right circular cylinder is calculated using the formula S = 2πr(h + r), where r is the radius and h is the height. This concept is crucial for understanding how the radius r(S) is derived from the surface area S, and it provides context for the relationship between the dimensions of the cylinder and its surface area.
Recommended video:
09:07Example 1: Minimizing Surface Area
Function Interpretation
Interpreting the result of a limit involves understanding what the limit signifies in the context of the problem. In this case, finding lim S→0^+ r(S) will reveal the radius of the cylinder as the surface area approaches zero, which can provide insights into the geometric implications of a cylinder with minimal surface area, such as its shape and dimensions.
Recommended video:
04:39Derivative of the Natural Logarithmic Function Example 7
Related Practice
Textbook Question
Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).
f(x)=√x^2+2x+6−3 / x−1
Textbook Question
Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x)=cos x+2√x / √x.
Textbook Question
Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).
f(x)=|1−x^2| / x(x+1)
Textbook Question
Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).
f(x)=3e^x+10 / e^x
1
views
Textbook Question
Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x)=|1−x^2| / x(x+1)
2
views