\lim _{x\to \:2}\left(x^2+4x+\frac{1}{2}\right)

To solve the limit:

\[
\lim_{x \to 2} \left( x^2 + 4x + \frac{1}{2} \right)
\]

Since this is a polynomial function with no undefined terms as \(x \to 2\), we can directly substitute \(x = 2\) into the expression:

\[
x^2 + 4x + \frac{1}{2} = (2)^2 + 4(2) + \frac{1}{2} = 4 + 8 + \frac{1}{2}
\]

Now, simplify:

\[
4 + 8 + \frac{1}{2} = 12 + \frac{1}{2} = \frac{24}{2} + \frac{1}{2} = \frac{25}{2}
\]

Thus, the limit is:

\[
\lim_{x \to 2} \left( x^2 + 4x + \frac{1}{2} \right) = \frac{25}{2}
\]

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

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Here are some related questions to explore:
1. What is the limit of \( x^2 + 4x + \frac{1}{2} \) as \( x \to -2 \)?
2. How would the result change if the function included a denominator that caused an indeterminate form?
3. What are the general rules for evaluating polynomial limits as \( x \to c \)?
4. Can limits be evaluated using numerical approximation methods?
5. How do limits behave when approaching infinity in similar functions?

**Tip:** When dealing with polynomial functions, directly substituting the value of \(x\) often works, as polynomials are continuous everywhere.

\lim _{x\to 2}\left(x^2+4x+\frac{1}{2}\right)

Learn how to find the limit of the polynomial function x^2 + 4x + 1/2 as x approaches 2 using substitution. This problem highlights key calculus concepts and applies the continuity theorem for polynomial functions. Suitable for high school students in Grades 10-12.

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