Evaluate the limits:
a) lim
𝑥𝑥→1+
4𝑥𝑥−3
√𝑥𝑥 −1

We are tasked with evaluating the following limit:

\[
\lim_{{x \to 1^+}} \frac{4x - 3}{\sqrt{x} - 1}
\]

### Step 1: Direct Substitution
First, let's check what happens if we directly substitute \( x = 1 \):

\[
\frac{4(1) - 3}{\sqrt{1} - 1} = \frac{4 - 3}{1 - 1} = \frac{1}{0}
\]

This results in a division by zero, which suggests that the limit may approach infinity, but we need to determine if it's positive or negative infinity, or if it does not exist.

### Step 2: Analyze the Behavior for \( x \to 1^+ \)
As \( x \to 1^+ \), the numerator \( 4x - 3 \) approaches \( 1 \) from the positive side because \( 4x - 3 = 4(1^+) - 3 \approx 1 \).

For the denominator \( \sqrt{x} - 1 \), we have \( \sqrt{x} \) approaching 1, but for values slightly greater than 1, \( \sqrt{x} > 1 \). Therefore, \( \sqrt{x} - 1 \) will be positive and small.

Thus, for \( x \to 1^+ \), the numerator approaches 1 and the denominator approaches a small positive value. Therefore, the whole fraction grows large and positive.

### Step 3: Conclusion
The limit tends towards positive infinity:

\[
\lim_{{x \to 1^+}} \frac{4x - 3}{\sqrt{x} - 1} = +\infty
\]

Would you like to see more details or have further questions about this solution?

Here are some related questions to explore:
1. What happens to the limit as \( x \to 1^- \) instead?
2. How would you evaluate limits involving higher powers of \( x \)?
3. Can you apply L'Hopital's rule to this problem?
4. What would the limit be if the numerator were \( 4x^2 - 3 \)?
5. How does the concept of one-sided limits differ from general limits?

**Tip:** When you encounter division by zero in limits, analyze the signs and growth of the numerator and denominator near the point of interest to determine the limit's behavior.

Evaluate the limits: a) lim x→1+ (4x - 3) / (√x - 1)

Understand how to evaluate the one-sided limit as x approaches 1 from the positive side for the function (4x - 3) / (√x - 1). This problem covers key calculus concepts like direct substitution, limits approaching infinity, and the behavior of functions near critical points. Suitable for students in Grades 11-12.

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