Evaluate the following limits :
`Lim_(x to 0) 5^(x) sin (a/(5^(x)))`

Answer

To evaluate the limit \( \lim_{x \to 0} 5^x \sin\left(\frac{a}{5^x}\right) \), we can follow these steps:

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

### Step 2: Substitute \( 5^x \) in the denominator
We can rewrite \( 5^x \) as \( \frac{1}{5^{-x}} \):
\[ y = \lim_{x \to 0} \frac{\sin\left(\frac{a}{5^x}\right)}{5^{-x}} \]

### Step 3: Multiply and divide by \( a \)
To use the standard limit \( \lim_{u \to 0} \frac{\sin u}{u} = 1 \), we multiply and divide by \( a \):
\[ y = \lim_{x \to 0} \frac{\sin\left(\frac{a}{5^x}\right)}{\frac{a}{5^x}} \cdot \frac{a}{5^{-x}} \]

### Step 4: Identify the limit of the sine term
As \( x \to 0 \), \( 5^x \to 1 \), hence \( \frac{a}{5^x} \to a \). Therefore:
\[ \lim_{x \to 0} \frac{\sin\left(\frac{a}{5^x}\right)}{\frac{a}{5^x}} = 1 \]

### Step 5: Evaluate the remaining limit
Now we evaluate the remaining limit:
\[ y = 1 \cdot \lim_{x \to 0} \frac{a}{5^{-x}} = 1 \cdot a \cdot \lim_{x \to 0} 5^x \]
Since \( 5^x \to 1 \) as \( x \to 0 \):
\[ y = a \cdot 1 = a \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} 5^x \sin\left(\frac{a}{5^x}\right) = a \]

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