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Chapter 12 Class 11 Limits and Derivatives | 5 questions | about 4 minutes

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Question 1 of 5
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Question 1 of 5
For what integers \( m \) and \( n \) does both \( \lim_{x \to 0} f(x) \) and \( \lim_{x \to 1} f(x) \) exist, given:
\( f(x) = \begin{cases} mx^2 + n, & x < 0 \\ mx + m, & 0 \le x \le 1 \\ mx^3 + n, & x > 1 \end{cases} \)
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Question 2 of 5
Evaluate \( \lim_{x \to 0} \frac{ax + x \cos x}{b \sin x} \) where \( b \ne 0 \).
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Question 3 of 5
Consider \( f(x) = \frac{1}{x^2} \) for positive real numbers \( x > 0 \). What behavior does \( f(x) \) exhibit as \( x \to 0^+ \)?
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Question 4 of 5
Evaluate \( \lim_{x \to a} \frac{x^{5/2} - a^{5/2}}{\sqrt{x} - \sqrt{a}} \) for \( a > 0 \).
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Question 5 of 5
Evaluate \( \lim_{x \to -1} \frac{x^{10} + x^5 + 1}{x - 1} \).
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