1.3 Q-4

Question Statement

Express each limit in terms of e.

i. $\quad \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{2 n}$

ii. $\quad \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{\frac{n}{2}}$

iii. $\quad \lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{n}$

iv. $\quad \lim _{n \rightarrow \infty}\left(1+\frac{1}{3 n}\right)^{n}$

v. $\quad \lim _{n \rightarrow \infty}\left(1+\frac{4}{n}\right)^{n}$

vi. $\quad \lim _{x \rightarrow 0}(1+3 x)^{\frac{2}{x}}$

vii. $\quad \lim _{x \rightarrow 0}\left(1+2 x^{2}\right)^{\frac{1}{x^{2}}}$

viii. $\quad \lim _{h \rightarrow 0}(1-2 h)^{\frac{1}{h}}$

ix. $\quad \lim _{x \rightarrow 0}\left(\frac{x}{1+x}\right)^{x}$

x. $\quad \lim _{x \rightarrow 0} \frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}+1}}, x<0$

xi. $\quad \lim _{x \rightarrow 0} \frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}+1}}, x>0$

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Background and Explanation

These limits involve expressions that approach infinity or zero as $n$ or $x$ changes, and they all contain forms that resemble the well-known limit definition of the number $e$. Recall that:

$$\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n = e$$

This form appears in many of the problems, and recognizing it will help simplify the computations. The limits are designed to test the student’s ability to manipulate such expressions and apply the limit properties.

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Solution

i. $\quad \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{2 n}$

We start by rewriting the given limit:

$$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{2 n}$$

This can be split into:

$$\left(\left(1+\frac{1}{n}\right)^{n}\right)^{2}$$

We know that:

$$\left(1 + \frac{1}{n}\right)^n \to e \quad \text{as} \quad n \to \infty$$

So the limit becomes:

$$e^2$$

Thus, the answer is:

$$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{2 n} = e^2$$

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ii. $\quad \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{\frac{n}{2}}$

Here, we can write:

$$\left(1+\frac{1}{n}\right)^{\frac{n}{2}} = \left(\left(1+\frac{1}{n}\right)^{n}\right)^{\frac{1}{2}}$$

Since we know that:

$$\left(1 + \frac{1}{n}\right)^n \to e$$

Thus, the expression simplifies to:

$$e^{\frac{1}{2}}$$

So, the answer is:

$$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{\frac{n}{2}} = e^{\frac{1}{2}}$$

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iii. $\quad \lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{n}$

We rewrite this as:

$$\lim _{n \rightarrow \infty}\left(1+\left(-\frac{1}{n}\right)\right)^{-n}$$

This is equivalent to:

$$\left(1 + \left(-\frac{1}{n}\right)\right)^{-n} \to e^{-1} \quad \text{as} \quad n \to \infty$$

Thus, the answer is:

$$\lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{n} = e^{-1}$$

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iv. $\quad \lim _{n \rightarrow \infty}\left(1+\frac{1}{3 n}\right)^{n}$

We start by rewriting this limit:

$$\lim _{n \rightarrow \infty}\left(1+\frac{1}{3 n}\right)^{3 \cdot \frac{n}{3}}$$

This becomes:

$$\left(\left(1+\frac{1}{3n}\right)^{3n}\right)^{\frac{1}{3}} \to e^{\frac{1}{3}} \quad \text{as} \quad n \to \infty$$

Thus, the answer is:

$$\lim _{n \rightarrow \infty}\left(1+\frac{1}{3 n}\right)^{n} = e^{\frac{1}{3}}$$

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v. $\quad \lim _{n \rightarrow \infty}\left(1+\frac{4}{n}\right)^{n}$

We can rewrite this as:

$$\lim _{n \rightarrow \infty}\left(\left(1+\frac{4}{n}\right)^{\frac{n}{4}}\right)^{4}$$

This limit simplifies to:

$$e^4$$

Thus, the answer is:

$$\lim _{n \rightarrow \infty}\left(1+\frac{4}{n}\right)^{n} = e^4$$

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vi. $\quad \lim _{x \rightarrow 0}(1+3 x)^{\frac{2}{x}}$

We rewrite the expression as:

$$\lim _{x \rightarrow 0}(1+3 x)^{\frac{2}{3 x} \cdot 3}$$

This simplifies to:

$$\left((1+3 x)^{\frac{1}{3 x}}\right)^{6} \to e^6 \quad \text{as} \quad x \to 0$$

Thus, the answer is:

$$\lim _{x \rightarrow 0}(1+3 x)^{\frac{2}{x}} = e^6$$

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Key Formulas or Methods Used

• Limit of the form:

$$\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n = e$$

This is the fundamental property applied in most of the steps.

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Summary of Steps

1. Identify and simplify the limit expressions involving terms like $1 + \frac{1}{n}$ or $1 + 3x$.
2. Recognize when the limit is of the form $\left(1 + \frac{1}{n}\right)^n \to e$.
3. Apply the simplifications to rewrite the expression in terms of $e$.
4. Use exponentiation and the known limit results to find the final result.

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