The expression in the image is:

$\lim_{{x \to 3}} \left(\frac{1}{x+1} - \frac{1}{4}\right)$

Let's solve this step by step.

Step 1: Evaluate the limit by substituting $x = 3$

Substitute $x = 3$ into the expression:

$\lim_{{x \to 3}} \left(\frac{1}{x+1} - \frac{1}{4}\right) = \frac{1}{3+1} - \frac{1}{4}$

Step 2: Simplify the expression

First, simplify the terms inside the limit:

$\frac{1}{3+1} = \frac{1}{4}$

So, the expression becomes:

$\frac{1}{4} - \frac{1}{4}$

Step 3: Subtract the fractions

Since the fractions have the same denominator, subtract the numerators:

$\frac{1-1}{4} = \frac{0}{4} = 0$

Final Answer

The limit is:

$\lim_{{x \to 3}} \left(\frac{1}{x+1} - \frac{1}{4}\right) = 0$

Would you like more details, or do you have any other questions?

Relative Questions:

1. What is the importance of evaluating limits in calculus?
2. How does the concept of continuity relate to limits?
3. Can limits be used to find the derivative of a function?
4. What happens if the limit involves an indeterminate form like $\frac{0}{0}$?
5. How do you apply L'Hôpital's rule to solve complex limits?

Tip:

When dealing with limits, always check if direct substitution is possible before considering more advanced techniques like factoring or L'Hôpital's rule.