The expression you've provided is:

$\lim_{{x \to 2}} \left( -3x^2 + 4x \right)$

Let's break it down and solve it step by step:

1. The limit notation means we want to evaluate the function $-3x^2 + 4x$ as $x$ approaches 2.
2. Since this is a polynomial, the limit can be evaluated directly by substituting $x = 2$ into the expression.

Now, let's substitute $x = 2$ into the function:

$-3(2)^2 + 4(2) = -3(4) + 8 = -12 + 8 = -4$

Thus, the limit is:

$\lim_{{x \to 2}} \left( -3x^2 + 4x \right) = -4$

Do you need further details or clarifications?

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Here are 5 related questions you might explore:

1. What is the limit of $-3x^2 + 4x$ as $x \to -2$?
2. How do you solve limits involving rational functions?
3. What is the difference between one-sided and two-sided limits?
4. Can you explain the limit laws that are applied when solving polynomial limits?
5. How do limits behave for functions with discontinuities?

Tip: For polynomial functions, limits can always be found by direct substitution if the function is continuous at the point of interest.