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The limit expression provided in the image is:

\[
\lim_{x \to 2} \left( \frac{4}{x - 4} + \frac{1}{2 - x} \right)
\]

### Step-by-Step Solution:

1. **Simplify the second fraction**: Notice that \( 2 - x \) is the negative of \( x - 2 \). So, we can rewrite \( \frac{1}{2 - x} \) as \( \frac{-1}{x - 2} \).

Thus, the expression becomes:

\[
\lim_{x \to 2} \left( \frac{4}{x - 4} - \frac{1}{x - 2} \right)
\]

2. **Factor and simplify**: Before substituting \( x = 2 \), observe that both terms have forms that will yield undefined results when directly substituting \( x = 2 \). So we can try factoring and simplifying to resolve this.

3. **Combine the fractions**: The least common denominator (LCD) of \( x - 4 \) and \( x - 2 \) is \( (x - 4)(x - 2) \). We can combine the two terms:

\[
\frac{4}{x - 4} - \frac{1}{x - 2} = \frac{4(x - 2) - 1(x - 4)}{(x - 4)(x - 2)}
\]

4. **Simplify the numerator**:

\[
4(x - 2) - 1(x - 4) = 4x - 8 - x + 4 = 3x - 4
\]

Thus, the expression becomes:

\[
\frac{3x - 4}{(x - 4)(x - 2)}
\]

5. **Evaluate the limit**: Now, substitute \( x = 2 \):

- The numerator becomes \( 3(2) - 4 = 6 - 4 = 2 \).
- The denominator becomes \( (2 - 4)(2 - 2) = (-2)(0) = 0 \).

Since the denominator is \( 0 \) and the numerator is nonzero, this means the limit does **not exist** in the standard sense; it results in a vertical asymptote, which leads to **infinity** or **negative infinity** depending on the direction of approach.

So, the limit does **not exist**.

Would you like further clarification or details? Here are some related questions to extend your understanding:

1. What are the conditions under which a limit tends to infinity?
2. How do you apply L'Hopital's Rule in indeterminate forms like \( \frac{0}{0} \)?
3. What is the geometric meaning of a vertical asymptote in limits?
4. Can the limit from the left side and the right side be different?
5. How do you solve limits involving complex rational expressions?

**Tip:** Always check for factorizations or common terms when facing undefined forms in limits!

The limit problem submitted by the user is: lim as x approaches 2 of (4/(x-4)) + (1/(2-x))

Learn how to solve the limit problem lim as x approaches 2 of (4/(x-4)) + (1/(2-x)). This problem involves simplifying rational expressions and identifying cases where the limit does not exist. Suitable for high school and college calculus students.

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