Find the limit as x approaches 2 of (4x² + 7x - 2)/(3x² + 8x + 2).

Step-by-step solution

1. 1

   Test the denominator at the target point before anything else. A rational function is continuous everywhere its denominator is nonzero, in which case the limit is simply the function value. So the first question is whether $3x^2+8x+2$ vanishes at $x=2$.

2. 2

   Evaluate the denominator. $3(2)^2+8(2)+2 = 12+16+2 = 30 \neq 0.$ Because it is nonzero, there is no indeterminate form - no factoring, conjugates or L'Hopital are needed.

3. 3

   Evaluate the numerator. $4(2)^2+7(2)-2 = 16+14-2 = 28.$

4. 4

   Form the quotient. $\lim_{x\to2}\frac{4x^2+7x-2}{3x^2+8x+2} = \frac{28}{30}.$

5. 5

   Reduce the fraction. $28$ and $30$ share the factor $2$, so $\frac{28}{30} = \frac{14}{15} \approx 0.9\overline{3}.$ Since $14 = 2\cdot7$ and $15 = 3\cdot5$ have no common factor, this is lowest terms.

6. 6

   Note where substitution would fail. The denominator's roots are $x = \tfrac{-8\pm\sqrt{64-24}}{6} = \tfrac{-4\pm\sqrt{10}}{3} \approx -0.2792$ and $-2.3874$. At either of those points the limit would require separate work; at $x=2$ the function is perfectly well behaved.