If `lim_(xto0)phi(x)=a^(2)` . where `ane0`. then what is `lim_(xto0)((x)/(a))` equal to ?

Options

A. `a^(2)`
B. `a^(-2)`
C. `-a^(2)`
D. `-a`

Answer

To solve the limit problem step by step, we start with the given information:

**Given:**
\[ \lim_{x \to 0} \phi(x) = a^2 \quad \text{where } a \neq 0 \]

**We need to find:**
\[ \lim_{x \to 0} \frac{x}{a} \]

### Step 1: Rewrite the limit
We can express the limit we want to find in terms of \(x\):
\[ \lim_{x \to 0} \frac{x}{a} = \frac{1}{a} \lim_{x \to 0} x \]

### Step 2: Evaluate the limit of \(x\)
As \(x\) approaches 0, the limit of \(x\) is straightforward:
\[ \lim_{x \to 0} x = 0 \]

### Step 3: Substitute back into the limit
Now substitute this result back into our expression:
\[ \lim_{x \to 0} \frac{x}{a} = \frac{1}{a} \cdot 0 = 0 \]

### Final Answer
Thus, we conclude that:
\[ \lim_{x \to 0} \frac{x}{a} = 0 \]

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