valuate the limit, if it exists. (If an answer does not exist, enter DNE.)
lim 
x→2 
x2 − 9x + 4
x − 2
 

To evaluate the limit 

\[
\lim_{x \to 2} \frac{x^2 - 9x + 4}{x - 2},
\]

we can start by checking whether substituting \( x = 2 \) directly into the function works. If direct substitution results in an indeterminate form (like \(\frac{0}{0}\)), we'll need to simplify the expression.

### Step 1: Direct substitution
Substitute \( x = 2 \) into the expression:

\[
\frac{(2)^2 - 9(2) + 4}{2 - 2} = \frac{4 - 18 + 4}{0} = \frac{-10}{0}.
\]

Since dividing by zero is undefined, direct substitution does not work. We do not get the indeterminate form \( \frac{0}{0} \); instead, we have \( \frac{-10}{0} \), which suggests the limit does not exist.

### Conclusion:
The limit is **DNE** (Does Not Exist), as the expression tends to infinity when \( x \to 2 \).

---

Would you like more details on how to handle different cases of limits approaching zero? 

Here are 5 related questions you might be interested in:
1. What is the limit of \( \frac{x^2 - 4}{x - 2} \) as \( x \to 2 \)?
2. How do you apply L'Hopital's Rule to limits that yield \( \frac{0}{0} \) indeterminate forms?
3. What happens when you encounter a limit that approaches infinity?
4. Can the limit of a rational function always be determined by direct substitution?
5. How can factoring help simplify expressions for limits?

**Tip**: When evaluating limits, always check for direct substitution first, and if it results in an indeterminate form, consider algebraic simplifications or applying limit rules!

Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.) lim x→2 (x^2 − 9x + 4) / (x − 2)

Learn how to evaluate the limit of the function (x^2 - 9x + 4)/(x - 2) as x approaches 2. This detailed explanation covers direct substitution, the behavior of rational functions near undefined points, and why this limit does not exist. Suitable for advanced high school and college students.

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