Textbook Question
Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.
Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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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 2.39
Estimate the following limits using graphs or tables.
lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4
Verified step by step guidance1
Identify the limit expression: \( \lim_{{x \to 1}} \frac{9(\sqrt{2x} - x^4 - 3\sqrt{x})}{1 - x^{3/4}} \).
Recognize that direct substitution of \( x = 1 \) results in an indeterminate form \( \frac{0}{0} \).
Consider using a table of values to estimate the limit by choosing values of \( x \) that approach 1 from both the left and the right.
Alternatively, graph the function \( f(x) = \frac{9(\sqrt{2x} - x^4 - 3\sqrt{x})}{1 - x^{3/4}} \) and observe the behavior as \( x \) approaches 1.
Analyze the behavior of the numerator and the denominator separately as \( x \to 1 \) to understand the limit's behavior.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. Evaluating limits can involve direct substitution, factoring, or using special techniques like L'Hôpital's rule when dealing with indeterminate forms.
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Continuity
Continuity refers to a property of functions where they do not have any abrupt changes, jumps, or holes at a given point. A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits, as discontinuities can lead to undefined or infinite limits.
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Graphical Analysis
Graphical analysis involves using the visual representation of a function to estimate limits and understand its behavior. By plotting the function, one can observe trends, identify asymptotes, and determine the value the function approaches as the input nears a specific point. This method is particularly useful for complex functions where algebraic manipulation may be challenging.
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Related Practice
Textbook Question
Evaluate each limit and justify your answer.
lim x→2 (3 / 2x^5−4x^2−50)^4
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Textbook Question
Use the definitions given in Exercise 57 to prove the following infinite limits.
lim x→1^+ 1 /1 − x=−∞
Textbook Question
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→3 −5x / √4x − 3
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Textbook Question
Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.
lim x→7 f(x)=9, where f(x)={3x−12 if x≤7
x+2 if x>7
Textbook Question
Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?
f(x)=(2x−3)^2/3