Evaluate the limit limx→2(−3(x−2)/4|x−2|).

We are asked to evaluate the limit:

\[
\lim_{x \to 2} \left( -\frac{3(x - 2)}{4 |x - 2|} \right)
\]

### Step 1: Break it into two cases

Since the expression involves \( |x - 2| \), we need to handle this piecewise, depending on whether \( x \) is approaching 2 from the left (i.e., \( x \to 2^- \)) or from the right (i.e., \( x \to 2^+ \)).

#### Case 1: \( x \to 2^+ \) (Approaching from the right)

When \( x > 2 \), we have \( |x - 2| = x - 2 \). Therefore, the expression becomes:

\[
\lim_{x \to 2^+} \left( -\frac{3(x - 2)}{4(x - 2)} \right)
\]

Simplifying this:

\[
\lim_{x \to 2^+} \left( -\frac{3}{4} \right) = -\frac{3}{4}
\]

#### Case 2: \( x \to 2^- \) (Approaching from the left)

When \( x < 2 \), we have \( |x - 2| = -(x - 2) \). Therefore, the expression becomes:

\[
\lim_{x \to 2^-} \left( -\frac{3(x - 2)}{4(-(x - 2))} \right)
\]

Simplifying this:

\[
\lim_{x \to 2^-} \left( \frac{3}{4} \right) = \frac{3}{4}
\]

### Step 2: Conclusion

We observe that the limit from the right is \( -\frac{3}{4} \) and the limit from the left is \( \frac{3}{4} \). Since these two one-sided limits are not equal, the two-sided limit does not exist.

Thus, the limit is:

\[
\lim_{x \to 2} \left( -\frac{3(x - 2)}{4 |x - 2|} \right) \text{ does not exist.}
\]

Would you like further details or have any questions?

Here are some related questions:
1. How does the absolute value function impact limits?
2. What are one-sided limits and why are they important?
3. Can a function have different one-sided limits?
4. How do you evaluate limits involving absolute value?
5. What is the significance of piecewise functions in limit evaluation?

**Tip:** When working with absolute values in limits, always consider both sides (left and right) to ensure the correct limit behavior.

Evaluate the limit limx→2(−3(x−2)/4|x−2|).

Explore how to solve the limit limx→2(−3(x−2)/4|x−2|) by breaking it down into one-sided limits. This problem involves key concepts like absolute value, piecewise functions, and limit evaluation, suitable for advanced high school or undergraduate calculus 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