lim->x-a- (absolute amount of x-1)/(x^2-a^2)

The expression you've written seems to represent a limit problem involving a one-sided limit. Let's clarify the expression to ensure we interpret it correctly:

\[
\lim_{x \to a^-} \frac{|x - 1|}{x^2 - a^2}
\]

This is the limit of the function \( \frac{|x - 1|}{x^2 - a^2} \) as \( x \) approaches \( a \) from the left (denoted by \( a^- \)).

### To solve this:
1. Factor the denominator:
   \[
   x^2 - a^2 = (x - a)(x + a)
   \]

2. Consider the behavior of the absolute value term \( |x - 1| \) in two cases based on whether \( x > 1 \) or \( x < 1 \).

#### Case 1: \( a > 1 \)
In this case, \( x - 1 \) is positive as \( x \to a^- \), so:
\[
|x - 1| = x - 1
\]

Thus, the expression becomes:
\[
\lim_{x \to a^-} \frac{x - 1}{(x - a)(x + a)}
\]

You can directly substitute \( x = a \) if the expression doesn't result in an indeterminate form. In this case, you can substitute:
\[
\frac{a - 1}{(a - a)(a + a)} = \frac{a - 1}{0}
\]

This leads to an undefined or infinite limit.

#### Case 2: \( a < 1 \)
In this case, \( x - 1 \) is negative as \( x \to a^- \), so:
\[
|x - 1| = -(x - 1)
\]

The expression then becomes:
\[
\lim_{x \to a^-} \frac{-(x - 1)}{(x - a)(x + a)}
\]
Again, when substituting \( x = a \), this will also lead to a similar undefined or infinite limit.

### Conclusion:
In both cases, the limit does not exist as it results in an undefined form. Therefore, the limit tends to infinity or does not exist, depending on the context and sign of the function.

Would you like to explore more details on how these limits behave for different values of \( a \)? 

Here are five related questions to deepen your understanding:
1. How does the sign of \( a \) affect the limit?
2. Can the expression be redefined to handle the case where \( a = 1 \)?
3. How would this limit change if the absolute value was removed?
4. What happens when \( a \to 1^+ \) instead of \( a^- \)?
5. Could we apply L'Hôpital's Rule to solve this limit?

**Tip:** When dealing with absolute values in limits, always break the problem into cases based on the sign of the expression inside the absolute value.

Explore the solution to the limit lim(x -> a^-) of the expression |x - 1| / (x^2 - a^2). This problem covers key concepts such as factoring difference of squares, analyzing absolute values, and evaluating one-sided limits. Suitable for college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation