Question

\(\lim _{x \rightarrow 0}\left\\{(1+x)^{\frac{2}{x}}\right\\}\) (where \(\\{x\\}\) denotes the fractional part of \(\mathrm{x}\) ) is equal to (A) \(\mathrm{e}^{2}-7\) (B) \(e^{2}-8\) (C) \(\mathrm{e}^{2}-6\) (D) None of these

Short answer

A) \(e^2\) B) \(\\{e^2\\}\) C) \(2\) D) None of these Answer: A) \(e^2\)

Step-by-step solution

Understanding the Fractional Part of x

The notation \(\\{x\\}\) refers to the fractional part of x, which is the part of x that remains after removing the integer part. Mathematically, it can be represented as \(\\{x\\} = x - \lfloor x \rfloor\), where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to x. Since we're considering the limit as x approaches 0, we can say that \(\\{x\\} \approx x\) (as x is very close to 0, the fractional part is virtually the same as x itself).

Substituting Fractional Part of x

Now that we understand the fractional part, we can make the substitution in the given expression: \((1+x)^{\frac{2}{x}} \approx(1+x)^{\frac{2}{\\{x\\}}}\).

Finding the Limit

Let's find the limit of the expression as x approaches 0. We will apply L'Hôpital's Rule to the given expression, which states that if the limit \(\lim_{x \rightarrow a} \frac{f(x)}{g(x)}\) exists, then \(\lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}\) also exists, and their values are equal. Before applying L'Hôpital's Rule, let's rewrite the expression: \(\lim_{x \rightarrow 0} (1+x)^{\frac{2}{x}} = e^{\lim_{x \rightarrow 0} \frac{2}{x} \ln(1+x)}\) Now, we need to find the limit of the exponent: \(\lim_{x \rightarrow 0} \frac{2}{x} \ln(1+x)\) Apply L'Hôpital's Rule: \(\lim_{x \rightarrow 0} \frac{\frac{2}{(1+x)}}{1} = \frac{2}{1} = 2\) Thus, the limit of the exponent is 2.

Final Answer

We found the limit of the exponent, so we can now find the limit of the expression: \(\lim_{x \rightarrow 0} (1+x)^{\frac{2}{x}} = e^{\lim_{x \rightarrow 0} \frac{2}{x} \ln(1+x)} = e^{2}\). Since our result is \(e^2\), which is not equal to any of the given options, the correct answer is: (D) None of these