Questions and Answers
What is lim $x→0$ (f(f(x)) closest to?
What is lim $x→1^-$ f(x) for the function f, defined as (x - 1)/(√(x) - 1) if x < 1, (x^2 + x - 2)/(x - 1) if x > 1?
Use the lim $x→0$ (sin x)/(x) to determine lim $x→0$ (x cos 5x)/(sin 5x). What is the limit?
Use the graph of g to determine lim $x→2^+$ g(x). What is the limit?
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What is lim $x→2$ (2f(x) - 3g(x)) given the values f(2) = 4, g(2) = 4, lim $x→2$ f(x) = 2, lim $x→2$ g(x) = 3?
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Study Notes
Limits and Functions Overview
- Limit Notation: lim x→c f(x) indicates the value that f(x) approaches as x approaches c.
Specific Limits Examples
- For lim x→0 (f(f(x))), the limit value is 3.
- The limit lim x→1- f(x) for defined function segments results in a value of 2 when evaluating the left-side function (x - 1)/(√(x) - 1) for x < 1 and the right-side function (x^2 + x - 2)/(x - 1) for x > 1.
Application of Trigonometric Limits
- To find lim x→0 (x cos 5x)/(sin 5x), use the known limit lim x→0 (sin x)/x, leading to the conclusion that the limit equals 1/5.
Evaluating Limits from Graphs
- Graph observation can help determine limits, such as lim x→2+ g(x), which is found to be 1.
Limit Evaluation from Tables
- Calculate limits from function values given in tables. For lim x→2 (2f(x) - 3g(x)), where f(2) = 4 and g(2) = 4, alongside lim x→2 f(x) = 2 and lim x→2 g(x) = 3, the resulting limit is -7.
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Description
This quiz focuses on determining limits using algebraic properties and manipulation, as covered in section 01.04. Test your understanding of key limit concepts and functions with these flashcards to prepare for your calculus exam.
