The Same Limits

$\displaystyle\lim_{x\rightarrow \infty} \frac{1}{x}=0$ is true as for larger and larger values of x , $1/x$ approaches 0 (we can see the graph of $1/x$ for more clarity).

But $\displaystyle\lim_{x\rightarrow 0} \frac{1}{x}=\infty$ is wrong because if we approach 0 from the left side, the values tend to $-\infty$ whereas if we approach 0 from the right side, the values tend to $+\infty$ . Since the right and left hand side limits are unequal the limit at 0 does not exist.