Let $$ f $$ be the function given by $$ f(x)=2 x \mathrm{e}^{2 x} $$ 1. Find $$ \lim _{x \rightarrow-\infty} f(x) $$ and $$ \lim _{x \rightarrow \infty} f(x) $$

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To evaluate the limits of the function $$ f(x) = 2x e^{2x} $$ as $$ x $$ approaches $$ -\infty $$ and $$ \infty $$, we'll analyze the behavior of each component of the function.

### 1. Limit as $$ x \to -\infty $$:

$$ \lim_{x \to -\infty} 2x e^{2x} $$

- **Behavior of $$ 2x $$ as $$ x \to -\infty $$:**
  - $$ 2x $$ decreases without bound (i.e., $$ 2x \to -\infty $$).

- **Behavior of $$ e^{2x} $$ as $$ x \to -\infty $$:**
  - $$ e^{2x} $$ approaches $$ 0 $$ because the exponential function decays rapidly as its exponent becomes large and negative.

- **Combined Behavior:**
  - The product $$ 2x e^{2x} $$ involves a term approaching $$ -\infty $$ multiplied by a term approaching $$ 0 $$.
  - However, the exponential decay of $$ e^{2x} $$ dominates the linear term $$ 2x $$. This means the entire expression approaches $$ 0 $$.

- **Conclusion:**
  $$ \lim_{x \to -\infty} 2x e^{2x} = 0 $$

### 2. Limit as $$ x \to \infty $$:

$$ \lim_{x \to \infty} 2x e^{2x} $$

- **Behavior of $$ 2x $$ as $$ x \to \infty $$:**
  - $$ 2x $$ increases without bound (i.e., $$ 2x \to \infty $$).

- **Behavior of $$ e^{2x} $$ as $$ x \to \infty $$:**
  - $$ e^{2x} $$ grows exponentially without bound.

- **Combined Behavior:**
  - Both $$ 2x $$ and $$ e^{2x} $$ are increasing towards $$ \infty $$.
  - The exponential growth of $$ e^{2x} $$ dominates the linear term $$ 2x $$, leading the entire expression to grow without bound.

- **Conclusion:**
  $$ \lim_{x \to \infty} 2x e^{2x} = \infty $$

### Final Answer:

$$
\lim_{x \to -\infty} f(x) = 0 \quad \text{and} \quad \lim_{x \to \infty} f(x) = +\infty
$$
$$ \lim_{x \to -\infty} f(x) = 0 \quad \text{and} \quad \lim_{x \to \infty} f(x) = +\infty $$

Let be the function given by

Find and

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1. Limit as :

2. Limit as :

Final Answer:

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