https://file.aiquickdraw.com/mathbot/file/17264676974222cn16a34.jpg

Let's analyze the two limit problems provided in the image.

### Problem (a):
\[
\lim_{x \to \frac{7}{4}^+} \frac{-14x}{7 - 4x}
\]
Here, we need to evaluate the limit as \(x\) approaches \(\frac{7}{4}\) from the right (\(x \to \frac{7}{4}^+\)).

1. **Substitute** \(x = \frac{7}{4}\) into the denominator:
   \[
   7 - 4x = 7 - 4 \times \frac{7}{4} = 7 - 7 = 0
   \]
   Since the denominator approaches 0, we need to check the sign of the denominator as \(x\) approaches \(\frac{7}{4}\) from the right.

   - If \(x > \frac{7}{4}\), then \(7 - 4x < 0\), meaning the denominator is negative.

2. **Analyze the numerator**:
   \[
   -14x \quad \text{as} \quad x \to \frac{7}{4}^+
   \]
   The numerator \( -14x \) is negative as well, since \( -14 \times \frac{7}{4} = -24.5 \).

3. **Behavior of the limit**:
   As \(x\) approaches \(\frac{7}{4}\) from the right, the numerator approaches a negative number, and the denominator approaches a small negative value (since \(7 - 4x\) becomes slightly negative). The overall fraction thus approaches positive infinity because the fraction is negative divided by negative:
   \[
   \lim_{x \to \frac{7}{4}^+} \frac{-14x}{7 - 4x} = +\infty
   \]

### Problem (b):
\[
\lim_{x \to \frac{7}{4}^-} \frac{-14x}{7 - 4x}
\]
This time, we are evaluating the limit as \(x\) approaches \(\frac{7}{4}\) from the left (\(x \to \frac{7}{4}^-\)).

1. **Substitute** \(x = \frac{7}{4}\) into the denominator (same as before):
   \[
   7 - 4x = 0
   \]
   As \(x\) approaches \(\frac{7}{4}\) from the left, \(7 - 4x > 0\), meaning the denominator is positive.

2. **Analyze the numerator**:
   Again, the numerator \( -14x \) is negative.

3. **Behavior of the limit**:
   As \(x\) approaches \(\frac{7}{4}\) from the left, the numerator is negative, and the denominator approaches a small positive value. This results in the fraction approaching negative infinity:
   \[
   \lim_{x \to \frac{7}{4}^-} \frac{-14x}{7 - 4x} = -\infty
   \]

### Final Answers:
- (a) \( \lim_{x \to \frac{7}{4}^+} \frac{-14x}{7 - 4x} = +\infty \)
- (b) \( \lim_{x \to \frac{7}{4}^-} \frac{-14x}{7 - 4x} = -\infty \)

Would you like further details or have any questions?

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Here are five related questions to further your understanding:
1. How does the behavior of the denominator influence the sign of the limit?
2. Why do we need to check the direction from which \(x\) approaches \(\frac{7}{4}\)?
3. Can you find another limit where the numerator and denominator both approach zero?
4. How would the limit change if the function were \(\frac{14x}{7 - 4x}\) instead of \(\frac{-14x}{7 - 4x}\)?
5. What happens to the limit if the numerator is constant instead of depending on \(x\)?

**Tip:** When solving limits involving fractions, always check the signs of both the numerator and denominator as \(x\) approaches the given value.

Evaluate the following limits: (a) lim_{x → 7/4^+} (-14x)/(7 - 4x), (b) lim_{x → 7/4^-} (-14x)/(7 - 4x).

Learn how to evaluate one-sided limits for the function (-14x)/(7 - 4x) as x approaches 7/4. This tutorial provides step-by-step instructions for calculating the limits from the left and right. Suitable for students studying calculus.

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