What is the limit $lim_{x \to 0} x^{2} sin (1/ x)$ ?

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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 $f (c) = 6$ ?