$$ \lim _ { t \rightarrow 0 } ( 8 \sin ( 10 t ) \ln ( 10 t ) ) = \square $$

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UpStudy AI · Accepted Answer

To evaluate the limit:

$$
\lim_{t \rightarrow 0} \left( 8 \sin(10t) \ln(10t) \right)
$$

let's break it down step by step.

### Step 1: Understand the Components
- As $$ t \rightarrow 0 $$, $$ 10t \rightarrow 0 $$.
- The sine function behaves as $$ \sin(10t) \approx 10t $$ for small $$ t $$.
- The natural logarithm $$ \ln(10t) $$ approaches $$ -\infty $$ as $$ t \rightarrow 0^+ $$ (since $$ \ln(0) $$ is undefined and tends to $$ -\infty $$ from the positive side).

### Step 2: Approximate the Expression
Using the small-angle approximation for sine:

$$
\sin(10t) \approx 10t
$$

Substituting this into the original expression:

$$
8 \sin(10t) \ln(10t) \approx 8 \cdot 10t \cdot \ln(10t) = 80t \ln(10t)
$$

### Step 3: Analyze the Limit of the Approximation
Consider the limit:

$$
\lim_{t \rightarrow 0^+} 80t \ln(10t)
$$

Notice that as $$ t \rightarrow 0^+ $$:
- $$ t $$ approaches $$ 0 $$.
- $$ \ln(10t) $$ approaches $$ -\infty $$.

However, the product $$ t \ln(10t) $$ tends to $$ 0 $$. This is because $$ t $$ approaches zero faster than $$ \ln(10t) $$ approaches $$ -\infty $$. Formally:

$$
\lim_{t \rightarrow 0^+} t \ln t = 0
$$

### Step 4: Conclusion
Since both components of the expression tend to zero, the overall limit is:

$$
\lim_{t \rightarrow 0} \left( 8 \sin(10t) \ln(10t) \right) = 0
$$

**Answer:** 0
The limit is 0.

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