Questions and Answers
Evaluate the limit as $x$ approaches 3 for the expression $\frac{x^2 - 9}{x^2 - 3x}$. What is the result?
What is the value of the limit as $x$ approaches -3 for $\frac{x^2 - 9}{1 - \cos x}$?
When evaluating $\lim_{x \to -\infty} \frac{x - 9x - 5}{3x^3 + x - 2}$, what is the result?
What is the result for $\lim_{t \to 0} \frac{t^2}{x^2 - 81}$?
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Determine if the function $f(x) = \begin{cases} \cos(x - 2) & \text{if } x \geq 2 \ 1 + x & \text{if } x < 2 \end{cases}$ is continuous at $x = 2$. What is the outcome?
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Identify the horizontal asymptote of the function $f(x) = \frac{x^4 - 8x^2 + 7}{1 + x}$. What is the result?
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When is the particle described by the motion equation $s(t) = t^3 - 13t^2 + 35t - 15$ at rest?
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For the function $f(x) = \frac{2x + 1}{x^2 - 5x + 6}$, what is the limit as $x$ approaches 2?
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Study Notes
Limit Evaluations
- For limit evaluations, key techniques involve direct substitution, factoring, and recognizing indeterminate forms.
- a) Evaluate (\lim_{x \to 3} \frac{x^2 - 9}{x^2 - 3x}) to find a finite value or determine if it diverges.
- b) Evaluate (\lim_{x \to -3^+} \frac{x^2 - 9}{1 - \cos x}) to analyze behavior approaching the limit.
- c) Assess (\lim_{x \to \frac{\pi}{2}} \frac{x \sin(4t)}{\tan(3t)}) to determine the limit's behavior as (x) approaches (\frac{\pi}{2}).
- d) Check (\lim_{t \to 0} \frac{t^2}{x^2 - 81}) for limit value evaluation.
- e) Analyze (\lim_{x \to 9} \frac{\sqrt{x} - 3}{x^2 - 3x^3 + 1}) for finite or infinite limits.
- f) Find (\lim_{x \to -\infty} \frac{x - 9x - 5}{3x^3 + x - 2}) to investigate limit behavior at negative infinity.
- g) Evaluate (\lim_{x \to -\infty} \sqrt{x^3 + 1 + x^6}) for limit trends as (x) approaches negative infinity.
Differentiation
- a) Differentiate (y = \frac{(2x + x^4) \cos x}{\sec x + x}) by applying quotient and product rules.
- b) Differentiate (y = \frac{\tan x - x}{3}) using the quotient rule.
- c) Differentiate (y = (3x^2 - \sin x + 1)(\csc x + \sqrt{x})) using product rule.
- d) Find derivative (y = \frac{e^x \cos x}{1 - \sqrt{x} x}) using the quotient and product rules.
- e) Differentiate (y = \frac{1}{2 + 3\sqrt{x}}) applying the chain rule as necessary.
Higher-Order Derivatives
- a) Find the second derivative (y = 3x^4 - \sqrt{x} + \pi - x).
- b) Differentiate (y = e^x \sin x) to investigate second derivatives.
Definition of the Derivative
- Use the limit definition of the derivative to derive (f'(x)) for (f(x) = 2x + 1).
Finding Horizontal Tangents
- Identify points where the derivative of (y = x^4 - 8x^2 + 7) equals zero to find horizontal tangents.
Tangent Lines
- Establish equation of the tangent line at the point ((-1, 0)) for the curve (f(x) = \frac{2x - 5x + 7}{x < 2}).
Continuity of Functions
- Determine continuity of piecewise function (f(x) = \begin{cases} \cos(x - 2) & \text{if } x \geq 2 \end{cases}) at (x = 2).
Asymptotic Behavior
- Evaluate limits as (x) approaches positive and negative infinities to identify horizontal and vertical asymptotes.
- Assess continuity and differentiability of the function across its domain.
Particle Motion Analysis
- Determine the velocity of the particle from the motion equation (s(t) = t^3 - 13t^2 + 35t - 15).
- Calculate velocity at specific time instances, analyze conditions for rest, and determine directions of movement.
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Description
Test your knowledge on evaluating limits and differentiation techniques with this quiz. Each question presents a different scenario to apply the fundamental concepts of calculus. Challenge yourself to find limits approaching finite and infinite values, along with differentiating complex functions.
