Panhellenic Examinations (Greece) · Mathematics

Limits, Continuity and Global Theorems

14 practice questions, each with the answer and an explanation.

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 14 questions

What is the value of $lim_{x \to 0} \frac{s i n ( 5 x )}{t a n ( 2 x )}$ ?
What is the limit $lim_{x \to + \infty} (x^{2} + 6 x - x)$ ?
Let $f : [1, 5] \to R$ be continuous with $f (1) = - 3$ and $f (5) = 7$ . Which statement is guaranteed by Bolzano's theorem?
What is the horizontal asymptote of the function $f (x) = \frac{4 e ^{x} + 3}{2 e ^{x} + 5}$ as $x \to + \infty$ ?
What is the limit $lim_{x \to 0} \frac{1 - c o s ( 4 x )}{x ^{2}}$ ?
If $f$ is continuous on $[0, 4]$ with $f (0) = 2$ and $f (4) = 10$ , which theorem guarantees a value $c \in (0, 4)$ where $f (c) = 6$ ?
What is the vertical asymptote of $f (x) = \frac{x ^{2} - 9}{x ^{2} - 5 x + 6}$ ?
What is the limit $lim_{x \to 1} \frac{l n x}{x ^{2} - 1}$ ?
Let $f : [a, b] \to R$ be continuous. The Extreme Value Theorem guarantees that:
Find the limit $lim_{x \to 0} \frac{e ^{3 x} - 1}{s i n ( 5 x )}$ .
What is the limit $lim_{x \to + \infty} \frac{3 x ^{3} - 5 x + 2}{7 x ^{3} + 2 x ^{2} - 1}$ ?
For what value of $k$ is the function $f (x) = {\frac{x ^{2} - 4}{x - 2}, k, x \neq = 2 x = 2$ continuous at $x = 2$ ?
What is the oblique (slant) asymptote of $f (x) = \frac{2 x ^{2} + 5 x - 1}{x + 1}$ as $x \to + \infty$ ?
What is the limit $lim_{x \to 0} x^{2} sin (1/ x)$ ?

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