To find the limit:

$\lim_{{x \to 1}} \frac{{[x]}}{x}$

Here, $[x]$ denotes the greatest integer function, or the floor function, which gives the largest integer less than or equal to $x$.

Step-by-step Solution:

1. Case 1: Approaching from the right ($x \to 1^+$):

   • If $x$ is slightly greater than 1, then $[x] = 1$.
   • The expression becomes: $\frac{1}{x}$
   • As $x \to 1^+$, $\frac{1}{x} \to \frac{1}{1} = 1$.

2. Case 2: Approaching from the left ($x \to 1^-$):

   • If $x$ is slightly less than 1, then $[x] = 0$.
   • The expression becomes: $\frac{0}{x} = 0$
   • As $x \to 1^-$, the expression evaluates to 0.

Conclusion:

The left-hand limit as $x \to 1^-$ is 0, and the right-hand limit as $x \to 1^+$ is 1. Since the left-hand and right-hand limits are not equal, the limit does not exist.

Final Answer:

$\lim_{{x \to 1}} \frac{{[x]}}{x} \text{ does not exist.}$

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the definition of the greatest integer function?
2. How do you compute limits with piecewise functions?
3. What are left-hand and right-hand limits?
4. How do you prove that a limit does not exist?
5. What is the relationship between continuity and limits?

Tip: Always check both left-hand and right-hand limits when solving limits involving piecewise or floor functions.