## Limits requiring a bit of thought

### Task number: 2986

Decide whether the following limits exist. If they do, determine their values.

#### Variant 1

\(\displaystyle \lim_{x\to 0} \frac{1}{x^2}. \)

#### Variant 2

\(\displaystyle \lim_{x\to 0} \frac{1}{x^3}. \)

#### Variant 3

\(\displaystyle \lim_{x \to \infty} \hbox{tan} \left( \frac{\pi}2 + \frac{1}{x^2 + 4}\right). \)

#### Variant 4

\(\displaystyle \lim_{x \to 0} \sin \left(\frac 1x\right). \)

#### Variant 5

\(\displaystyle \lim_{x \to 0} x \sin \left(\frac 1x\right). \)

#### Variant 6

\(\displaystyle \lim_{x \to 0} \frac{\sin\left(x \sin \left(\frac 1x\right)\right)}{x \sin \left(\frac 1x\right)}. \)