lim
x→1

root(3 + f(x))
given f(x)=1 at 1

To evaluate the limit \(\lim_{x \to 1} \sqrt{3 + f(x)}\) given that \(f(x) = 1\) at \(x = 1\), we follow these steps:

### Step 1: Substitute the given value
Since \(f(x) = 1\) at \(x = 1\), we can substitute this directly into the expression inside the square root.

\[
\lim_{x \to 1} \sqrt{3 + f(x)} = \sqrt{3 + f(1)}
\]

### Step 2: Evaluate the expression
Substitute \(f(1) = 1\) into the equation:

\[
\sqrt{3 + 1} = \sqrt{4}
\]

### Step 3: Simplify
\(\sqrt{4} = 2\).

### Final Answer:
\[
\lim_{x \to 1} \sqrt{3 + f(x)} = 2
\]

Would you like more details on this process or have any other questions?

Here are 5 related questions for further exploration:
1. What is the definition of a limit?
2. How does the continuity of \(f(x)\) affect the evaluation of limits?
3. What happens if \(f(x)\) was not continuous at \(x = 1\)?
4. Can you explain how to evaluate limits involving more complex functions?
5. How would you find the limit if \(f(x)\) were given as a piecewise function?

**Tip:** When dealing with limits, always consider substituting directly if the function is continuous at the point of interest.

Learn how to evaluate the limit \(\lim_{x \to 1} \sqrt{3 + f(x)}\) given that \(f(x) = 1\) at \(x = 1\). Follow a step-by-step solution to understand the process involving square roots and function evaluation. Suitable for advanced high school or college students.

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