h. lim
𝑥→ 1
𝑓(𝑥)𝑖𝑓 𝑓 (𝑥) = {
𝑥
+
3
𝑖𝑓 𝑥 < 1
2
𝑥
𝑖𝑓 𝑥 = 1
√
5
𝑥
− 1
𝑖𝑓 𝑥 > 1 (Provide a tables of values) 

To find the limit of \( f(x) \) as \( x \) approaches 1, we'll need to evaluate the left-hand limit (\( \lim_{x \to 1^-} f(x) \)) and the right-hand limit (\( \lim_{x \to 1^+} f(x) \)). Here's the piecewise function defined:

\[
f(x) =
\begin{cases}
x + 3 & \text{if } x < 1 \\
2x & \text{if } x = 1 \\
\sqrt{5x - 1} & \text{if } x > 1
\end{cases}
\]

### Left-hand Limit (\( x \to 1^- \))
For \( x < 1 \), \( f(x) = x + 3 \). As \( x \) approaches 1 from the left:
\[
\lim_{x \to 1^-} f(x) = 1 + 3 = 4
\]

### Right-hand Limit (\( x \to 1^+ \))
For \( x > 1 \), \( f(x) = \sqrt{5x - 1} \). As \( x \) approaches 1 from the right:
\[
\lim_{x \to 1^+} f(x) = \sqrt{5(1) - 1} = \sqrt{4} = 2
\]

### Table of Values Around \( x = 1 \)
Let's evaluate \( f(x) \) for values around 1 to observe the behavior:

| \( x \)  | \( f(x) \) if \( x < 1 \)  | \( f(x) \) at \( x = 1 \) | \( f(x) \) if \( x > 1 \)  |
|----------|----------------------------|---------------------------|---------------------------|
| 0.9      | \( 0.9 + 3 = 3.9 \)         | -                         | -                         |
| 0.99     | \( 0.99 + 3 = 3.99 \)       | -                         | -                         |
| 0.999    | \( 0.999 + 3 = 3.999 \)     | -                         | -                         |
| 1        | -                           | \( 2(1) = 2 \)            | -                         |
| 1.001    | -                           | -                         | \( \sqrt{5(1.001) - 1} \approx 2.002 \) |
| 1.01     | -                           | -                         | \( \sqrt{5(1.01) - 1} \approx 2.02 \)  |
| 1.1      | -                           | -                         | \( \sqrt{5(1.1) - 1} \approx 2.19 \)  |

### Conclusion
- The left-hand limit as \( x \) approaches 1 is 4.
- The right-hand limit as \( x \) approaches 1 is 2.
- Since the left-hand limit and the right-hand limit are not equal, the limit of \( f(x) \) as \( x \) approaches 1 does not exist.

Would you like more details or have any questions?

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**5 Related Questions:**
1. What is the importance of checking both left-hand and right-hand limits?
2. How do you determine the continuity of a piecewise function at a point?
3. Can the limit exist if the left-hand and right-hand limits are different?
4. How do you compute the limit when one side approaches infinity?
5. What are some techniques to find the limit of more complex piecewise functions?

**Tip:** Always check both left-hand and right-hand limits to determine if the overall limit exists at a point.

h. lim x → 1 f(x) if f(x) = { x + 3 if x < 1, 2x if x = 1, √(5x - 1) if x > 1 (Provide a table of values)

Explore the limit of a piecewise function where f(x) includes linear terms and a square root function. Learn how to evaluate the left-hand and right-hand limits as x approaches 1, along with a detailed table of values. This problem focuses on limits, continuity, and piecewise functions. Suitable for Grades 10-12.

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