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 2.32
Evaluate each limit and justify your answer.
lim x→2 (3 / 2x^5−4x^2−50)^4
Verified step by step guidance1
Identify the type of limit problem: This is a limit of a function as x approaches a specific value.
Substitute x = 2 into the function to check if it results in an indeterminate form or a defined value.
Calculate the expression inside the parentheses: \(2x^5 - 4x^2 - 50\) by substituting x = 2.
Evaluate the expression \(3 / (2x^5 - 4x^2 - 50)\) using the result from the previous step.
Raise the result to the power of 4 to find the limit.
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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 specific points, which is crucial for evaluating expressions that may not be directly computable at those points.
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One-Sided Limits
Continuous Functions
A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits, as discontinuities can lead to undefined expressions or different limit values.
Recommended video:
05:34Intro to Continuity
Polynomial Functions
Polynomial functions are expressions that involve variables raised to whole number powers and are combined using addition, subtraction, and multiplication. In the given limit, the polynomial in the denominator can affect the limit's value, especially if it approaches zero, leading to potential indeterminate forms.
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6:04Introduction to Polynomial Functions
Related Practice
Textbook Question
Estimate the following limits using graphs or tables.
lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4
Textbook Question
Which one of the following intervals is not symmetric about x=5?
a.(1, 9)
b.(4, 6)
c.(3, 8)
d.(4.5, 5.5)
4
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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 x − 3 /|x − 3|
6
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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
Determine the following limits.