`lim_(xrarr0)(sin5x)/(x)` is equal to

Options

A. `5^(2)`
B. `(1)/(5^(2))`
C. 5
D. `1//5`

Answer

To solve the limit \( \lim_{x \to 0} \frac{\sin(5x)}{x} \), we can follow these steps:

### Step 1: Rewrite the limit
We start by rewriting the limit:
\[ y = \lim_{x \to 0} \frac{\sin(5x)}{x} \]

### Step 2: Multiply and divide by 5
Next, we multiply and divide the expression by 5:
\[ y = \lim_{x \to 0} \frac{\sin(5x)}{x} \cdot \frac{5}{5} = \lim_{x \to 0} \frac{5 \sin(5x)}{5x} \]

### Step 3: Factor out the constant
Now we can factor out the constant 5 from the limit:
\[ y = 5 \cdot \lim_{x \to 0} \frac{\sin(5x)}{5x} \]

### Step 4: Use the standard limit
We know from the standard limit that:
\[ \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \]
In our case, let \( u = 5x \). As \( x \to 0 \), \( u \to 0 \) as well. Thus:
\[ \lim_{x \to 0} \frac{\sin(5x)}{5x} = 1 \]

### Step 5: Substitute back into the equation
Substituting this result back into our equation gives us:
\[ y = 5 \cdot 1 = 5 \]

### Conclusion
Thus, the limit is:
\[ \lim_{x \to 0} \frac{\sin(5x)}{x} = 5 \]

### Final Answer
The final answer is:
\[ \boxed{5} \]