**Determine the limit (answer as appropriate, with a number,**

∞

**, −**

∞

**or does not exist).**
****

lim

*x*

→

2

*x* − 92

*x*2 − 4*x* + 4

**=**

We are asked to determine the limit:

\[
\lim_{x \to 2} \frac{x - 92}{x^2 - 4x + 4}
\]

### Step 1: Simplify the denominator
The denominator is \( x^2 - 4x + 4 \), which is a perfect square:
\[
x^2 - 4x + 4 = (x - 2)^2
\]

Thus, the expression becomes:
\[
\frac{x - 92}{(x - 2)^2}
\]

### Step 2: Substitution of \( x = 2 \)
Substituting \( x = 2 \) directly into the expression:

- The numerator becomes \( 2 - 92 = -90 \).
- The denominator becomes \( (2 - 2)^2 = 0^2 = 0 \).

This results in the form \( \frac{-90}{0} \), which indicates that the limit involves division by zero. 

### Step 3: Determine the behavior around \( x = 2 \)
Since the denominator approaches zero as \( x \to 2 \), we must analyze the sign of the expression near \( x = 2 \).

- When \( x \) is slightly greater than 2, say \( x = 2^+ \), \( (x - 2)^2 \) remains positive, but the numerator \( x - 92 \) is negative.
- When \( x \) is slightly less than 2, say \( x = 2^- \), the same applies because \( (x - 2)^2 \) is always positive, while \( x - 92 \) remains negative.

### Step 4: Conclusion
Since the numerator is negative and the denominator approaches 0 from the positive side, the fraction approaches negative infinity:

\[
\lim_{x \to 2} \frac{x - 92}{(x - 2)^2} = -\infty
\]

The limit is \( -\infty \).

Would you like more details or have any questions?

Here are five related questions for further exploration:
1. What happens if the limit is taken as \( x \to -2 \) for a similar expression?
2. How do you evaluate limits that result in the form \( \frac{0}{0} \)?
3. How does L'Hopital's Rule apply to indeterminate forms?
4. What are the different types of discontinuities in rational functions?
5. How do limits relate to the concept of derivatives in calculus?

**Tip:** Always check if the denominator can be factored or simplified when dealing with limits involving rational expressions.

Determine the limit (answer as appropriate, with a number, ∞, −∞ or does not exist). lim x → 2 (x − 92) / (x^2 − 4x + 4) =

Explore how to determine the limit of (x − 92) / (x^2 − 4x + 4) as x approaches 2. This problem involves limits, rational functions, and indeterminate forms. Detailed steps and analysis show why the limit is −∞, with key concepts like factoring quadratics and asymptotic behavior.

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