Construct a table and find the indicated limit.
If ​h(x)equals
StartFraction StartRoot x EndRoot plus 2 Over x minus 2 EndFraction​, 
then find ModifyingBelow lim With x right arrow 2 h left parenthesis x right parenthesis
.
Question content area bottom
Part 1
Complete the table below.
x
	
1.9
1.99
1.999
2.001
2.01
2.1
h left parenthesis x right parenthesis
enter your response here
enter your response here
enter your response here
enter your response here
enter your response here
enter your response here
​(Type integers or decimals rounded to four decimal places as​ needed.)

We are asked to compute the limit \( \lim_{{x \to 2}} h(x) \), where the function \( h(x) \) is given by:

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

To analyze this limit, we will evaluate the function at values of \( x \) that approach 2 from both the left and the right. These values are given in the table: \( 1.9, 1.99, 1.999, 2.001, 2.01, 2.1 \).

### Step-by-Step Evaluation

For each value of \( x \), we will compute \( h(x) \) by substituting \( x \) into the formula and rounding the result to four decimal places.

Let's compute the values for each point:

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

- For \( x = 1.9 \):
  \[
  h(1.9) = \frac{\sqrt{1.9} + 2}{1.9 - 2} = \frac{1.3784 + 2}{-0.1} = \frac{3.3784}{-0.1} = -33.784
  \]

- For \( x = 1.99 \):
  \[
  h(1.99) = \frac{\sqrt{1.99} + 2}{1.99 - 2} = \frac{1.4107 + 2}{-0.01} = \frac{3.4107}{-0.01} = -341.07
  \]

- For \( x = 1.999 \):
  \[
  h(1.999) = \frac{\sqrt{1.999} + 2}{1.999 - 2} = \frac{1.4141 + 2}{-0.001} = \frac{3.4141}{-0.001} = -3414.1
  \]

- For \( x = 2.001 \):
  \[
  h(2.001) = \frac{\sqrt{2.001} + 2}{2.001 - 2} = \frac{1.4142 + 2}{0.001} = \frac{3.4142}{0.001} = 3414.2
  \]

- For \( x = 2.01 \):
  \[
  h(2.01) = \frac{\sqrt{2.01} + 2}{2.01 - 2} = \frac{1.4177 + 2}{0.01} = \frac{3.4177}{0.01} = 341.77
  \]

- For \( x = 2.1 \):
  \[
  h(2.1) = \frac{\sqrt{2.1} + 2}{2.1 - 2} = \frac{1.4491 + 2}{0.1} = \frac{3.4491}{0.1} = 34.491
  \]

### Constructed Table

Now that we have computed the values, the table looks like this:

\[
\begin{array}{|c|c|}
\hline
x & h(x) \\
\hline
1.9 & -33.784 \\
1.99 & -341.07 \\
1.999 & -3414.1 \\
2.001 & 3414.2 \\
2.01 & 341.77 \\
2.1 & 34.491 \\
\hline
\end{array}
\]

### Conclusion

As \( x \) approaches 2 from both sides, \( h(x) \) becomes increasingly large in magnitude but with opposite signs on either side of \( x = 2 \). Therefore, we conclude:

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

This is because the function approaches negative infinity from the left and positive infinity from the right.

---

Would you like further details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. What is the general method for computing limits using a table of values?
2. How can you determine whether a limit exists if the function's values approach infinity?
3. What is the significance of one-sided limits in this problem?
4. Can you simplify the given function to make the limit more apparent?
5. How can L'Hôpital's Rule help in finding limits that involve indeterminate forms?

**Tip**: When analyzing limits numerically, always check the behavior from both sides of the target point (left and right).

Construct a table and find the indicated limit. If h(x) = (sqrt(x) + 2) / (x - 2), then find lim as x approaches 2 of h(x). Complete the table below.
x: 1.9, 1.99, 1.999, 2.001, 2.01, 2.1
h(x): (enter values rounded to four decimal places).

Learn how to find the limit of h(x) = (sqrt(x) + 2) / (x - 2) as x approaches 2, with a detailed table and step-by-step calculations. Analyze indeterminate forms and discover why the limit does not exist, suitable for students in Grades 10-12.

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