`lim_(xrarr0) x^2sin.(pi)/(x)`, is

Options

A. 1
B. 0
C. non-existant
D. `oo`

Answer

To solve the limit \( \lim_{x \to 0} x^2 \sin\left(\frac{\pi}{x}\right) \), we can follow these steps:

### Step 1: Rewrite the Limit
We start with the expression:
\[ \lim_{x \to 0} x^2 \sin\left(\frac{\pi}{x}\right) \]

### Step 2: Multiply and Divide by \(\pi\)
To simplify the expression, we multiply and divide by \(\pi\):
\[ \lim_{x \to 0} x^2 \sin\left(\frac{\pi}{x}\right) = \lim_{x \to 0} x^2 \cdot \frac{\sin\left(\frac{\pi}{x}\right)}{\frac{\pi}{x}} \cdot \frac{\pi}{x} \]
This can be rewritten as:
\[ \lim_{x \to 0} \left( x \cdot \pi \cdot \frac{\sin\left(\frac{\pi}{x}\right)}{\frac{\pi}{x}} \right) \]

### Step 3: Analyze the Limit of \(\frac{\sin\left(\frac{\pi}{x}\right)}{\frac{\pi}{x}}\)
As \( x \to 0 \), \(\frac{\pi}{x} \to \infty\). We know from the limit property that:
\[ \lim_{y \to \infty} \frac{\sin(y)}{y} = 0 \]
Thus, we can conclude:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{\pi}{x}\right)}{\frac{\pi}{x}} = 1 \]

### Step 4: Substitute Back into the Limit
Now we substitute this back into our limit:
\[ \lim_{x \to 0} \left( x \cdot \pi \cdot 1 \right) = \lim_{x \to 0} \pi x \]

### Step 5: Evaluate the Final Limit
As \( x \to 0 \), \( \pi x \to 0 \):
\[ \lim_{x \to 0} \pi x = 0 \]

### Conclusion
Thus, the final result is:
\[ \lim_{x \to 0} x^2 \sin\left(\frac{\pi}{x}\right) = 0 \]