Question

Calculator limits Use a calculator to approximate the following limits. $$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}$$

Short answer

Question: Using a calculator, approximate the value of the limit \(\lim_{n\to\infty} (1+\frac{1}{n})^n\). Answer: The approximate value of the given limit is \(\lim_{n\to\infty} (1+\frac{1}{n})^n \approx 2.7183\) for n = 1,000,000.

Step-by-step solution

Have a calculator ready

First, make sure you have a calculator that can handle large numbers and exponentiation. Most calculators, including the ones found on smartphones, will suffice.

Substitute large values for n

We will approximate the limit by plugging in large values for n. The greater the value of n, the closer we will get to the actual limit. Let's try the following values for n: - n = 100 - n = 10,000 - n = 1,000,000

Compute the expression for each n value

For each value of n, compute the expression \((1+\frac{1}{n})^n\). Make sure you are evaluating the expression correctly, calculating the fraction separately and then raising the result to the power of n: - For n = 100: \((1+\frac{1}{100})^{100} \approx 2.7048\) - For n = 10,000: \((1+\frac{1}{10000})^{10000} \approx 2.7181\) - For n = 1,000,000: \((1+\frac{1}{1000000})^{1000000} \approx 2.7183\)

Compare the results

As we increase the value of n, our approximation of the limit gets closer to the actual value. The more precise value of the limit is \(e \approx 2.718281828459045\): - Limit approximation for n = 100: \(2.7048\) - Limit approximation for n = 10,000: \(2.7181\) - Limit approximation for n = 1,000,000: \(2.7183\) So, using a calculator, we can approximate the given limit to be \(\lim_{n\to\infty} (1+\frac{1}{n})^n \approx 2.7183\) for n = 1,000,000.

Key concepts

Limits and Continuity

Understanding the concept of limits is essential in the realm of calculus, as it deals with values that are approaching a particular point, rather than just the values at that point. Consider a function that gets closer to a certain value as the input approaches some value; this 'certain value' is known as the limit of the function at that point.

For instance, when we observe the expression \(1+\frac{1}{n})^n\), as \(n\) becomes very large (approaching infinity), the expression's value approaches a specific constant. This process of reaching a value is what we call a limit. Continuity, on the other hand, implies that a function does not have abrupt changes in value— it flows smoothly. This is closely linked to the concept of limits, as a function that has a limit at every point within an interval and equals the function's value at those points is continuous in that interval.

Exponential Growth

Exponential growth occurs when a quantity increases by a constant proportion (rate) in a consistent time period, leading to a growth pattern that accelerates over time. In mathematical terms, this is often represented by functions of the form \(y = a\cdot e^{rt}\), where \(a\) is the initial amount, \(r\) is the growth rate, and \(t\) is time.

The constant \(e\), which is approximately equal to 2.71828, is the base of the natural logarithm and it's a key player in exponential functions. These functions model diverse real-world processes, including population growth, radioactive decay, and interest compounding. The properties of exponential functions make them valuable for describing scenarios where things grow very rapidly.

e (Mathematical Constant)

The number \(e\), known as Euler's number or the base of the natural logarithm, is a fundamental mathematical constant approximately equal to 2.718281828459045. This transcendental number is irrational, meaning it has an infinite non-repeating decimal expansion. It arises in many areas of mathematics, not just in exponential growth, but also in the study of compound interest, probability theory, and in the calculus concept of the natural logarithm.

In the example we are discussing, \(e\) is the basis of the exponential expression \(1+\frac{1}{n})^n\), as \(n\) approaches infinity. The understanding of \(e\) could be likened to \(\pi\), which is a ratio constant for circles— just like \(\pi\) is fundamental in geometry, \(e\) is fundamental in many calculus equations, including the natural exponential function \(e^x\).

Numerical Approximation

Numerical approximation methods are used to estimate values that might be too complicated or impossible to calculate exactly. In our exercise, we utilize a calculator to approximate the value of the limit as \(n\) approaches infinity, which is a practical application of numerical approximation.

As we input successively larger values for \(n\), the calculated result gets closer to the true limit value. This technique is not just limited to limits; it's used in many fields to approximate solutions to complex equations, perform integrations that can't be expressed in a simple formula, or in any situation where an exact answer is unattainable or impractical to reach. It is crucial to note that while numerical approximation provides us with an estimated value, understanding the accuracy and potential error in this estimate is also part of the process.