d) \( \lim _{x \rightarrow \frac{2}{3}} \frac{3 x^{2}+2 x-5}{x^{2}-x+5}= \)

To find the limit \( \lim _{x \rightarrow \frac{2}{3}} \frac{3 x^{2}+2 x-5}{x^{2}-x+5} \), we can start by substituting \( x = \frac{2}{3} \) directly into the function. Calculating the numerator: \[ 3 \left( \frac{2}{3} \right)^{2} + 2 \left( \frac{2}{3} \right) - 5 = 3 \cdot \frac{4}{9} + \frac{4}{3} - 5 = \frac{12}{9} + \frac{12}{9} - \frac{45}{9} = \frac{24 - 45}{9} = \frac{-21}{9} = -\frac{7}{3} \] Calculating the denominator: \[ \left( \frac{2}{3} \right)^{2} - \frac{2}{3} + 5 = \frac{4}{9} - \frac{2}{3} + 5 = \frac{4}{9} - \frac{6}{9} + \frac{45}{9} = \frac{4 - 6 + 45}{9} = \frac{43}{9} \] Now we can see that the limit becomes: \[ \frac{-\frac{7}{3}}{\frac{43}{9}} = -\frac{7}{3} \cdot \frac{9}{43} = -\frac{21}{43} \] Thus, the limit is: \[ \lim _{x \rightarrow \frac{2}{3}} \frac{3 x^{2}+2 x-5}{x^{2}-x+5} = -\frac{21}{43}. \]