Question

1. \\( \lim _ { x \

ightarrow 3 } \frac { x ^ { 4 } - 24 x ^ { 3 } + 206 x ^ { 2 } - 744 x + 945 } { x ^ { 3 } - 2 x ^ { 2 } - 7 } \\)

Explanation:

Step1: Substitute \(x = 3\) into the numerator

$$\begin{align*} &3^{4}-24\times3^{3}+206\times3^{2}-744\times3 + 945\\ =&81-24\times27+206\times9-2232 + 945\\ =&81-648+1854-2232 + 945\\ =&(81 + 1854+945)-(648 + 2232)\\ =&2880 - 2880\\ =&0 \end{align*}$$

Step2: Substitute \(x = 3\) into the denominator

$$\begin{align*} &3^{3}-2\times3^{2}-7\\ =&27-2\times9-7\\ =&27 - 18-7\\ =&2 \end{align*}$$

Step3: Calculate the limit

Since \(\lim_{x
ightarrow3}(x^{4}-24x^{3}+206x^{2}-744x + 945)=0\) and \(\lim_{x
ightarrow3}(x^{3}-2x^{2}-7)=2
eq0\), then \(\lim_{x
ightarrow3}\frac{x^{4}-24x^{3}+206x^{2}-744x + 945}{x^{3}-2x^{2}-7}=\frac{0}{2}\)

Answer:

\(0\)