Evaluating a polynomial limit using limit laws

Because $f(x) = 3x^2 - 5x + 2$ is a polynomial, it is continuous at every real number, including $x = 2$. Using the sum and difference rules, $\lim_{x\to 2} f(x) = 3\lim_{x\to2}x^2 - 5\lim_{x\to2}x + \lim_{x\to2}2$. The power rule gives $\lim_{x\to2}x^2 = \left(\lim_{x\to2}x\right)^2 = 2^2 = 4$, and $\lim_{x\to2}x = 2$.

Substituting these values in: $3(4) - 5(2) + 2 = 12 - 10 + 2 = 4$. This matches what direct substitution of $x=2$ into $f(x)$ would give, confirming the result. Therefore $\lim_{x\to2}(3x^2 - 5x + 2) = 4$.