The concept of limits is a fundamental aspect of calculus, particularly when dealing with exponential functions. One common problem arises when evaluating the limit of the form 1 * x^n as x approaches a. In this article, we will explore the limit of (1 cdot x^n) as x approaches a, and how it relates to the base of the natural logarithm, e, approximately equal to 2.718. We will also delve into the continuity of functions and the binomial theorem to provide a comprehensive understanding of this topic.

Introduction to Limits and Continuity

To begin, let's define the limit of a function. A limit describes the behavior of a function as the input values approach a certain point. Specifically, for the limit lim_{x to a} f(x) L to exist, the function values, f(x), must get arbitrarily close to a single value, L, as x gets arbitrarily close to a. If the function is continuous at a, then the limit as x approaches a is simply the value of the function at that point, i.e., lim_{x to a} f(x) f(a) .

The Limit of 1 * x^n as x Approaches a

Consider the limit of the function f(x) 1 cdot x^n as x approaches a. Since (1 cdot x^n) is a polynomial function and polynomials are continuous everywhere, we can directly substitute a into the function to find the limit:

lim_{x to a} (1 cdot x^n) 1 cdot a^n a^n

This direct substitution is valid because the function is continuous at every point in its domain.

Relating to the Base of the Natural Logarithm, e

The more complex aspect comes into play when we consider the limit of the form (e^{na}), where e is the base of the natural logarithm, approximately 2.718. This form is related to the exponential function f(x) e^x. The exponential function can be expressed using the limit definition of e:

e lim_{n to infty} left(1 frac{1}{n}right)^n

Using this definition and the binomial theorem, we can analyze more complex exponential functions. For example, if we are considering the expression (e^{na}), we can use the expansion of the binomial theorem:

(1 frac{1}{n})^n approx 1 n cdot frac{1}{n} frac{n(n-1)}{2!} cdot left(frac{1}{n}right)^2 cdots

Simplifying this, we get:

(1 frac{1}{n})^n approx 1 1 frac{1}{2!} cdots frac{1}{n^{n-1}}

This approximation helps us understand how the function behaves as n approaches infinity, and how it relates to the base of the natural logarithm, e.

Proving the Limit Using the Definition of e

To prove that lim_{x to a} e^{na} e^{na} , we need to use the definition of e and the properties of limits. Recall that:

e lim_{n to infty} left(1 frac{1}{n}right)^n

Now, consider the function g(n) (1 frac{a}{n})^n. We want to show that:

lim_{n to infty} g(n) e^a

Using the binomial theorem, we expand g(n):

g(n) left(1 frac{a}{n}right)^n 1 n cdot frac{a}{n} frac{n(n-1)}{2!} cdot left(frac{a}{n}right)^2 cdots left(frac{a}{n}right)^n

Simplifying, we get:

g(n) 1 a frac{a^2}{2!} cdots left(frac{a}{n}right)^n

As n approaches infinity, the term left(frac{a}{n}right)^n approaches 0, and the remaining terms form the infinite series that defines e^a. Therefore:

lim_{n to infty} g(n) e^a

Conclusion

In summary, the limit of 1 * x^n as x approaches a is a^n, which is straightforward due to the continuity of the polynomial function. However, for the more complex form of (e^{na}), we can use the limit definition of e and the binomial theorem to derive its behavior. Understanding these concepts is crucial for mastering advanced topics in calculus and real analysis.