\lim_{x\rarr2} $$$$ \frac{x+2}{\left(x-2\right)^2}=

To find the limit

\[
\lim_{x \to 2} \frac{x+2}{(x-2)^2}
\]

we can start by analyzing the behavior of the numerator and the denominator as $ x \to 2 $.

1. **Substitute $ x = 2 $:**
   - In the numerator, $ x + 2 = 2 + 2 = 4 $.
   - In the denominator, $ (x - 2)^2 = (2 - 2)^2 = 0 $.

This gives us a form of $ \frac{4}{0} $, which indicates a division by zero, suggesting that the limit may approach infinity or negative infinity. We need to examine the direction of approach for $ x \to 2 $.

2. **Analyze the sign as $ x \to 2^+ $ and $ x \to 2^- $:**
   - When $ x \to 2^+ $ (approaching from the right), $ x - 2 $ is a small positive number, so $ (x - 2)^2 $ is a small positive value.
   - When $ x \to 2^- $ (approaching from the left), $ x - 2 $ is a small negative number, but $ (x - 2)^2 $ is still a small positive value (since it’s squared).

In both cases, the denominator approaches $ 0^+ $, a small positive value, while the numerator remains close to $ 4 $. Therefore:

\[
\lim_{x \to 2} \frac{x+2}{(x-2)^2} = +\infty
\]

because the fraction becomes very large as $ x $ gets closer to 2 from either side.

### Final Answer
\[
\lim_{x \to 2} \frac{x+2}{(x-2)^2} = +\infty
\]

Would you like more details on this solution, or do you have further questions?

---

1. What is an example of a limit approaching infinity with a different denominator?
2. How does the square in $ (x - 2)^2 $ affect the behavior of this limit?
3. How would the result change if the numerator were $ x - 2 $ instead?
4. What happens if you take a similar limit but with $ (x - 2)^3 $ in the denominator?
5. Can limits approach negative infinity? Under what conditions?

**Tip:** When encountering a zero in the denominator, consider approaching from both sides to see if the limit goes to positive or negative infinity.

Learn how to evaluate the limit \lim_{x \to 2} \frac{x+2}{(x-2)^2} as it approaches infinity. This calculus problem demonstrates limit analysis with division by zero, explaining the behavior of functions as x approaches a specific value. Ideal for Grade 12 or college-level calculus students.

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