Question content area top
Part 1
Construct a table and find the indicated limit.
If ​h(x)equals=StartFraction StartRoot x EndRoot plus 2 Over x minus 2 EndFraction
x+2
x−2​, then find ModifyingBelow lim With x right arrow 2 h left parenthesis x right parenthesislimx→2h(x).
Question content area bottom
Part 1
Complete the table below.
x
1.91.9
1.991.99
1.9991.999
2.0012.001
2.012.01
2.12.1

We are asked to evaluate the limit of the function

\[
h(x) = \frac{\sqrt{x} + 2}{x - 2}
\]

as \( x \to 2 \).

### Part 1: Understanding the Problem
We are trying to find \(\lim_{x \to 2} h(x)\). Direct substitution of \(x = 2\) into \(h(x)\) leads to an indeterminate form, because the denominator becomes 0. Therefore, we'll use a table of values to explore the behavior of the function as \(x\) approaches 2 from both sides.

### Part 2: Constructing a Table
We will evaluate \(h(x)\) for values of \(x\) approaching 2 from both the left (values less than 2) and the right (values greater than 2). The values we will use are: 1.9, 1.99, 1.999, 2.001, 2.01, 2.1.

Let's now calculate \(h(x)\) for each value of \(x\).Here are the calculated values of \( h(x) \) for the given values of \( x \):

| \( x \)   | \( h(x) \)                |
|-----------|---------------------------|
| 1.9       | -33.784                   |
| 1.99      | -341.067                  |
| 1.999     | -3413.860                 |
| 2.001     | 3414.567                  |
| 2.01      | 341.774                   |
| 2.1       | 34.491                    |

### Part 3: Analyzing the Table
As \( x \) approaches 2 from the left (values less than 2), \( h(x) \) becomes increasingly negative. As \( x \) approaches 2 from the right (values greater than 2), \( h(x) \) becomes increasingly positive. This suggests that the function does not settle on a specific value as \( x \to 2 \); instead, the function tends toward \(\infty\) (positive or negative) as \( x \to 2\).

Thus, we conclude:

\[
\lim_{x \to 2} h(x) \text{ does not exist} 
\]

because the left-hand limit and right-hand limit do not agree.

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Would you like further details or have any questions?

Here are 5 related questions for practice:
1. What is the definition of a limit?
2. How can limits be used to identify vertical asymptotes?
3. What does it mean if a limit results in an indeterminate form?
4. What is the difference between one-sided and two-sided limits?
5. How can you use a table of values to estimate a limit?

**Tip:** When evaluating limits involving square roots and rational functions, always check if direct substitution leads to an indeterminate form before exploring other methods.

Construct a table and find the indicated limit. If h(x) = (√x + 2) / (x - 2), then find lim x→2 h(x). Complete the table below with values for x: 1.9, 1.99, 1.999, 2.001, 2.01, and 2.1.

Solve the limit lim x→2 h(x) where h(x) = (√x + 2) / (x - 2) using a table of values. Understand the behavior of the function as x approaches 2 and analyze indeterminate forms. Suitable for advanced high school students in Grades 10-12.

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