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Questions and Answers
What is lim $x→0$ (f(f(x)) closest to?
What is lim $x→0$ (f(f(x)) closest to?
- 1
- 2
- 3 (correct)
- 4
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?
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?
- Undefined
- 3
- 0
- 2 (correct)
Use the lim $x→0$ (sin x)/(x) to determine lim $x→0$ (x cos 5x)/(sin 5x). What is the limit?
Use the lim $x→0$ (sin x)/(x) to determine lim $x→0$ (x cos 5x)/(sin 5x). What is the limit?
- 1/5 (correct)
- 5
- 0
- 1
Use the graph of g to determine lim $x→2^+$ g(x). 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?
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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