Questions and Answers
For how many positive values of b does ( \lim_{x \to b} f(x) = 2 )?
Which of the following is the best estimate for the speed of the particle at time t=8?
At which value of t would the speed of the rocket most likely be greatest based on the data in the table?
If ( y = x(t) ) is a linear function, which of the following would give the best estimate of the speed of the particle at time t=20 minutes?
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Which of the following equations expresses that ( f(x) ) can be made arbitrarily close to 2 by taking ( x ) sufficiently close to 0?
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What must be true if the values of f(x) get closer to 7 as x gets closer to 4?
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Which of the following could be the graph of f if ( \lim_{x \to 1} f(x) = 3 )?
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Which of the following expressions equals 2?
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What is ( \lim_{x \to 5} f(x) )?
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Based on the data in the table, what is the best approximation for ( \lim_{x \to 3} f(x) )?
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Which conclusion is supported by the data in the table regarding ( \lim_{x o 4^+} f(x) )?
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Which of the following statements must be true about ( \lim_{x o 1} f(x) )?
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What is ( \lim_{x o 1^-} f(x) )?
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What is ( \lim_{x o 4} f(x) + 7g(x) )?
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What is ( \lim_{x o 0} (\cos(x) + 3e^{x^2}) )?
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What is ( \lim_{x o 9} f(x) ) if ( f(x) = \frac{x - 9}{\sqrt{x} - 3} )?
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What is ( \lim_{x o \frac{\pi}{2}} f(x) ) if ( f(x) = sin(x) - \frac{1}{cos^2(x)} )?
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Study Notes
Limit and Continuity in Functions
- Function Behaviour: f(x) = 0.1x^4 - 0.5x^3 - 3.3x^2 + 7.7x - 1.99 approaches a limit of 2 at three positive values of b.
- Particle Motion: At time t = 8, a particle's speed is estimated to be 0 based on its position graph.
- Rocket Height: The rocket's speed is likely greatest at t = 400 seconds based on given height data over time.
Estimating Speed and Limits
- Linear Function Speed: For a particle moving right, the best estimate of speed at t = 20 minutes is the slope of y = x(t).
- Limit Property: The function f(x) = (e^(2x) - 1)/x approaches 2 as x approaches 0, showing a specific limit property.
- Function Limits: If f(x) approaches 7 as x nears 4, then lim(x→4) f(x) = 7 holds true.
Graphical Interpretation
- Function Graphs: For lim(x→1) f(x) = 3, specific graph shapes are possible.
- Limit Evaluation: Expressions can often simplify to constants like 2 based on the function's graphical behaviour.
Continuous Functions and Data Interpretation
- Continuous Function Limits: Approximating lim(x→3) from a table of values gives an estimate of 5.
- Right-Hand Limit: Limit from the right at x = 4 indicates a value of 6 based on tabulated data.
- Definitive Conclusions: Data may not conclusively determine lim(x→1) f(x) based on the given values.
Calculating Limits
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Specific Function Limits:
- lim(x→1−) f(x) evaluates to 4.
- The combined limit of two functions at x = 4 results in 2.
- Cosine and exponential function limit as x approaches 0 calculates to 2.
Functional Transformations
- Function Transformation Limit: The limit involving f(x) = (x - 9)/(sqrt(x) - 3) transforms effectively as x approaches 9, focusing on the square root component.
- Complex Limit Evaluation: A limit problem involving polynomials approaches a value of 3 as x tends to 0.
Trigonometric Limits
- Trigonometric Functions: The evaluation lim(x→π/2) of f(x) = sin(x) - 1/cos^2(x) simplifies to an expression involving sin(x) as x approaches π/2 from the left.
Studying That Suits You
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Description
Test your understanding of key concepts in Unit 1 with this multiple-choice question quiz. This quiz covers functions and particle motion, presenting scenarios for estimation and limit evaluation. Perfect for reviewing material before exams.
