\lim\limits_{x\rightarrow 2}(2x^{2}-7x+4)

To solve the limit \(\lim\limits_{x\rightarrow 2}(2x^{2}-7x+4)\), we can directly substitute \(x = 2\) into the polynomial expression since it is continuous at \(x = 2\).

Let's compute:

\[
\lim\limits_{x\rightarrow 2}(2x^{2}-7x+4) = 2(2)^2 - 7(2) + 4
\]

Calculating each term:

\[
2(2)^2 = 2 \times 4 = 8
\]
\[
7(2) = 14
\]

Now, substituting these values back:

\[
8 - 14 + 4 = -6 + 4 = -2
\]

So, the limit is:

\[
\lim\limits_{x\rightarrow 2}(2x^{2}-7x+4) = -2
\]

Would you like more details or have any questions?

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Here are five related questions:
1. What is the limit \(\lim\limits_{x\rightarrow 3}(3x^{2}-5x+2)\)?
2. How do you find the limit of a rational function?
3. What is the significance of continuity when evaluating limits?
4. How can limits be used to find derivatives?
5. What is the definition of a limit at infinity?

**Tip:** When a function is continuous at a point, the limit at that point can be found simply by substituting the value into the function.

Learn how to evaluate the limit lim(x -> 2) (2x^2 - 7x + 4) with a detailed step-by-step solution. This problem involves polynomial functions and the concept of limits in calculus. Suitable for high school students.

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