Answer:
[tex]-5[/tex]
Explanation:
Here, we want to find the limit shown
To do this, we start by factorizing the numerator
Thus, we have that as:
[tex]\begin{gathered} 3t^2-7t\text{ + 2 = 3t}^2-6t-t+2 \\ =\text{ 3t\lparen t-2\rparen-1\lparen t-2\rparen = \lparen3t-1\rparen\lparen t-2\rparen} \end{gathered}[/tex]
Now, from here, we have the original fraction reduced to the following:
[tex]\begin{gathered} \frac{3t^2-7t+2}{2-t}\text{ = }\frac{(3t-1)(t-2)}{2-t}\text{ = }\frac{-(2-t)(3t-1)}{2-t} \\ \\ =\text{ -\lparen3t-1\rparen = 1-3t} \end{gathered}[/tex]
Finally, we find the limit as follows:
[tex]\lim_{t\to2}\text{ }\frac{3t^2-7t+2}{t-2}\text{ = }\lim_{t\to2}\text{ 1-3t = 1-3\lparen2\rparen = 1-6 = -5}[/tex]
