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.

---

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.