The given expression is:

$\lim_{{x \to 3}} \frac{1}{{x - 3}}$

Now, let's analyze the limit as $x$ approaches 3.

Step-by-Step Solution:

• The denominator becomes $x - 3$.
• As $x \to 3$, the expression becomes $\frac{1}{0}$, which is undefined.
• This implies the function approaches infinity or negative infinity depending on the direction from which $x$ approaches 3.

Directional limits:

1. From the left ($x \to 3^-$):

   • If $x \to 3$ from the left (values slightly less than 3), $x - 3$ will be negative.
   • Therefore, $\frac{1}{{x - 3}}$ tends towards $-\infty$.

2. From the right ($x \to 3^+$):

   • If $x \to 3$ from the right (values slightly greater than 3), $x - 3$ will be positive.
   • Therefore, $\frac{1}{{x - 3}}$ tends towards $+\infty$.

Conclusion:

The limit does not exist in the conventional sense, but we can say:

$\lim_{{x \to 3^-}} \frac{1}{{x - 3}} = -\infty$ $\lim_{{x \to 3^+}} \frac{1}{{x - 3}} = +\infty$

Would you like more details or have any further questions?

Here are 5 related questions:

1. What is the difference between a two-sided limit and a one-sided limit?
2. How do we handle limits involving infinity?
3. Can limits exist if one side goes to positive infinity and the other to negative infinity?
4. How would the limit change if the denominator was $(x - 3)^2$?
5. How can limits help in understanding discontinuities in a function?

Tip: When evaluating limits, always check both left-hand and right-hand behavior for points where the denominator can approach zero!