limx→5+1+2x−10−−−−−−√x+3

To evaluate the limit

\[
\lim_{{x \to 5^+}} \frac{1 + 2x - 10}{\sqrt{x + 3}},
\]

we will first simplify the expression in the numerator and then assess the behavior of both the numerator and the denominator as \(x\) approaches 5 from the right.

### Step 1: Simplify the Expression

First, let's simplify the numerator:

\[
1 + 2x - 10 = 2x - 9.
\]

Thus, the expression becomes:

\[
\frac{2x - 9}{\sqrt{x + 3}}.
\]

### Step 2: Evaluate the Limit

Now, substitute \(x = 5\) into the numerator and the denominator:

- **Numerator:** \(2(5) - 9 = 10 - 9 = 1\).
- **Denominator:** \(\sqrt{5 + 3} = \sqrt{8}\).

So, the limit can now be written as:

\[
\lim_{{x \to 5^+}} \frac{2x - 9}{\sqrt{x + 3}} = \frac{1}{\sqrt{8}}.
\]

### Step 3: Simplify the Result

Simplify the square root:

\[
\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}.
\]

Therefore, the limit is:

\[
\lim_{{x \to 5^+}} \frac{2x - 9}{\sqrt{x + 3}} = \frac{1}{2\sqrt{2}}.
\]

Alternatively, this can also be written as:

\[
\frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}.
\]

So, the final answer is:

\[
\lim_{{x \to 5^+}} \frac{1 + 2x - 10}{\sqrt{x + 3}} = \frac{\sqrt{2}}{4}.
\]

Would you like more details on any specific step or have any questions?

### Related Questions:

1. How do you find the limit of a function as \(x\) approaches a certain value?
2. What are the common techniques to simplify square roots?
3. How do you evaluate one-sided limits?
4. What is the difference between a one-sided limit and a two-sided limit?
5. How can you rewrite \(\frac{\sqrt{2}}{4}\) in a different form?

**Tip:** When evaluating limits involving square roots, simplifying the expressions beforehand often makes the calculation much easier!

Learn how to evaluate the limit lim(x -> 5+) (1 + 2x - 10) / sqrt(x + 3) step-by-step with a detailed solution. This problem covers the evaluation of limits involving square roots and algebraic simplification. Suitable for advanced high school and college-level students.

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