Evhate the sollowine limits (a) $$ \lim _{x \rightarrow 2}\left(x^{2}+x\right)\left(3 x^{2}+6\right) $$ (b) $$ \lim _{x \rightarrow-2} \sqrt{x^{2}+3 x+6} $$

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Certainly! Let's evaluate each of the given limits step by step.

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### Problem (a)

$$
\lim_{x \rightarrow 2} \left(x^{2} + x\right)\left(3x^{2} + 6\right)
$$

**Step 1: Direct Substitution**

Since the functions involved are polynomials, they are continuous everywhere. This means we can directly substitute $$ x = 2 $$ into the expression.

**Step 2: Substitute $$ x = 2 $$**

$$
\begin{align*}
\left(2^{2} + 2\right)\left(3 \times 2^{2} + 6\right) &= \left(4 + 2\right)\left(3 \times 4 + 6\right) \
&= (6)(12 + 6) \
&= 6 \times 18 \
&= 108
\end{align*}
$$

**Answer:**
$$
\lim_{x \rightarrow 2} \left(x^{2} + x\right)\left(3x^{2} + 6\right) = 108
$$

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### Problem (b)

$$
\lim_{x \rightarrow -2} \sqrt{x^{2} + 3x + 6}
$$

**Step 1: Direct Substitution**

First, check if direct substitution is possible by plugging $$ x = -2 $$ into the expression inside the square root.

$$
x^{2} + 3x + 6 = (-2)^{2} + 3(-2) + 6 = 4 - 6 + 6 = 4
$$

Since the expression inside the square root is 4, which is non-negative, the square root is defined.

**Step 2: Calculate the Square Root**

$$
\sqrt{4} = 2
$$

**Answer:**
$$
\lim_{x \rightarrow -2} \sqrt{x^{2} + 3x + 6} = 2
$$

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**Summary:**

- **Problem (a)** evaluates to **108**.
- **Problem (b)** evaluates to **2**.

If you have any further questions or need additional explanations, feel free to ask!
The limits are: (a) 108 (b) 2

Evhate the sollowine limits
(a)
(b)

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