Evaluate the following limits :
`Lim_( x to 0) ((1+5x^(2))/(1+3x^(2)))^(x^(1/2))`

Answer

To evaluate the limit

\[ \lim_{x \to 0} \left(\frac{1 + 5x^2}{1 + 3x^2}\right)^{\sqrt{x}}, \]

we can follow these steps:

### Step 1: Rewrite the Limit
We start by rewriting the limit expression:

\[ L = \lim_{x \to 0} \left(\frac{1 + 5x^2}{1 + 3x^2}\right)^{\sqrt{x}}. \]

### Step 2: Simplify the Fraction
As \(x\) approaches \(0\), both the numerator and denominator approach \(1\). Therefore, we can simplify the fraction:

\[ \frac{1 + 5x^2}{1 + 3x^2} \approx 1 + (5x^2 - 3x^2) = 1 + 2x^2. \]

### Step 3: Substitute the Simplified Fraction
Now we substitute this back into the limit:

\[ L = \lim_{x \to 0} \left(1 + 2x^2\right)^{\sqrt{x}}. \]

### Step 4: Use the Exponential Limit
We can use the fact that \((1 + u)^{v} \approx e^{uv}\) as \(u \to 0\). Here, \(u = 2x^2\) and \(v = \sqrt{x}\):

\[ L = \lim_{x \to 0} e^{\sqrt{x} \cdot 2x^2} = \lim_{x \to 0} e^{2x^{2.5}}. \]

### Step 5: Evaluate the Exponential Limit
As \(x\) approaches \(0\), \(2x^{2.5}\) approaches \(0\):

\[ L = e^0 = 1. \]

### Final Result
Thus, the limit evaluates to:

\[ \boxed{1}. \]

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