Short Quiz

Question 1 of 5

Evaluate \( \lim_{x \to 0} \frac{1 - \cos 2x}{x^2} \).

A \( 0 \) B \( 1 \) C \( 2 \) D \( 4 \)

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Question 2 of 5

Evaluate \( \lim_{x \to 0} \frac{\sin 4x}{\sin 2x} \).

A \( 1 \) B \( 2 \) C \( 4 \) D \( \frac{1}{2} \)

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Question 3 of 5

Consider the tangent construction diagram below. If the tangent line to \( y = f(x) \) at point \( P(a, f(a)) \) makes an angle \( \psi \) with the positive x-axis, what is the exact mathematical relation for derivative \( f'(a) \)?

A \( f'(a) = \sin \psi \) B \( f'(a) = \cos \psi \) C \( f'(a) = \tan \psi \) D \( f'(a) = \sec \psi \)

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Question 4 of 5

Consider \( f(x) = \frac{1}{x^2} \) for positive real numbers \( x > 0 \). What behavior does \( f(x) \) exhibit as \( x \to 0^+ \)?

A Approaches 0 B Approaches 1 C Grows arbitrarily large (\( +\infty \)) D Oscillates between -1 and +1

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Question 5 of 5

Find \( \lim_{x \to 5} f(x) \) where \( f(x) = |x| - 5 \).

A \( 5 \) B \( -5 \) C \( 0 \) D Does not exist

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