Find (A) the leading term of the polynomial, (B) the limit as $$ x $$ approaches $$ \infty $$, alld (C) the limit as $$ x $$ approaches - $$ \infty $$. $$ p(x)=19+3 x^{2}-6 x^{3} $$ (A) The leading term is $$ -6 x^{3} $$. (B) The limit of $$ p(x) $$ as $$ x $$ approaches $$ \infty $$ is $$ \square $$.

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Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{x\rightarrow -\infty}\left(19+3x^{2}-6x^{3}\right)$$
- step1: Rewrite the expression:
  $$\lim _{x\rightarrow -\infty}\left(19\right)+\lim _{x\rightarrow -\infty}\left(3x^{2}\right)-\lim _{x\rightarrow -\infty}\left(6x^{3}\right)$$
- step2: Calculate:
  $$19+\left(+\infty\right)-\left(-\infty\right)$$
- step3: Calculate:
  $$\left(+\infty\right)-\left(-\infty\right)$$
- step4: Calculate:
  $$+\infty$$
Calculate the limit $$ \lim_{x\rightarrow \infty} (19 + 3x^{2} - 6x^{3}) $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{x\rightarrow +\infty}\left(19+3x^{2}-6x^{3}\right)$$
- step1: Rewrite the expression:
  $$\lim _{x\rightarrow +\infty}\left(x^{3}\left(\frac{19}{x^{3}}+\frac{3}{x}-6\right)\right)$$
- step2: Rewrite the expression:
  $$\lim _{x\rightarrow +\infty}\left(x^{3}\right)\times \lim _{x\rightarrow +\infty}\left(\frac{19}{x^{3}}+\frac{3}{x}-6\right)$$
- step3: Calculate:
  $$\left(+\infty\right)\times \lim _{x\rightarrow +\infty}\left(\frac{19}{x^{3}}+\frac{3}{x}-6\right)$$
- step4: Calculate:
  $$\left(+\infty\right)\left(-6\right)$$
- step5: Rewrite the expression:
  $$-\infty$$
Let's break down the problem step by step:

### (A) Leading Term
The leading term of the polynomial $$ p(x) = 19 + 3x^{2} - 6x^{3} $$ is the term with the highest degree. In this case, the term with the highest degree is $$ -6x^{3} $$.

**Answer:** The leading term is $$ -6x^{3} $$.

### (B) Limit as $$ x $$ approaches $$ \infty $$
To find the limit of $$ p(x) $$ as $$ x $$ approaches $$ \infty $$, we observe the leading term, which dominates the behavior of the polynomial for large values of $$ x $$.

Calculating the limit:
$$
\lim_{x \rightarrow \infty} (19 + 3x^{2} - 6x^{3}) = -\infty
$$

**Answer:** The limit of $$ p(x) $$ as $$ x $$ approaches $$ \infty $$ is $$ -\infty $$.

### (C) Limit as $$ x $$ approaches $$ -\infty $$
Similarly, for $$ x $$ approaching $$ -\infty $$, the leading term $$ -6x^{3} $$ will again dominate the polynomial.

Calculating the limit:
$$
\lim_{x \rightarrow -\infty} (19 + 3x^{2} - 6x^{3}) = +\infty
$$

**Answer:** The limit of $$ p(x) $$ as $$ x $$ approaches $$ -\infty $$ is $$ +\infty $$.

### Summary of Answers:
- (A) The leading term is $$ -6x^{3} $$.
- (B) The limit as $$ x $$ approaches $$ \infty $$ is $$ -\infty $$.
- (C) The limit as $$ x $$ approaches $$ -\infty $$ is $$ +\infty $$.
(A) The leading term is $$ -6x^{3} $$. (B) The limit of $$ p(x) $$ as $$ x $$ approaches $$ \infty $$ is $$ -\infty $$. (C) The limit of $$ p(x) $$ as $$ x $$ approaches $$ -\infty $$ is $$ +\infty $$.

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