Proving the Limit of (e^x - 1) / x as x Approaches 0

The limit of (e^x - 1) / x as x approaches 0 is a fundamental concept in calculus and is often used in the definition of the derivative of the exponential function. This limit is equal to 1, and there are several methods to prove it. Here, we will explore the derivative definition, L'H?pital's rule, and a direct algebraic approach.

Derivative Definition Method

Let's begin with the derivative definition approach. The derivative of a function f(x) at x 0 is given by:

f'(0) limx→0 [f(x) - f(0)] / (x - 0)

In our case, let's take the function f(x) e^x. Thus:

f'(0) limx→0 [e^x - e^0] / x limx→0 (e^x - 1) / x

Given that the derivative of e^x is e^x, we have:

f'(0) e^0 1

Hence, we can conclude that:

limx→0 (e^x - 1) / x 1

L'H?pital's Rule Approach

L'H?pital's rule is a powerful tool for evaluating limits of the form (0/0) or (∞/∞). In our case, the limit is in the form (0/0) as x approaches 0:

limx→0 (e^x - 1) / x

By L'H?pital's rule, we can differentiate the numerator and the denominator with respect to x:

limx→0 [d/dx (e^x - 1)] / [d/dx x] limx→0 (e^x) / 1 limx→0 e^x e^0 1

Thus, we have:

limx→0 (e^x - 1) / x 1

Algebraic Approach

Another way to approach this problem is through algebraic manipulation. We can rewrite the expression as:

(e^x - 1) / x 1

This implies:

e^x - 1 x

To verify, let's substitute x 0:

e^0 - 1 0

1 - 1 0

Also, substituting x 1:

e^1 - 1 e^0

e - 1 1

In both cases, the equation holds true.

Taylor Series Expansion Approach

Finally, we can use the Taylor series expansion of e^x to prove this limit. The Taylor series expansion of e^x around x 0 is:

e^x 1 x (x^2)/2! (x^3)/3! ...

Subtracting 1 from both sides, we get:

e^x - 1 x (x^2)/2! (x^3)/3! ...

Now, consider the limit:

limx→0 (e^x - 1) / x limx→0 [x (x^2)/2! (x^3)/3! ...] / x

Dividing each term by x, we get:

limx→0 [1 (x)/2! (x^2)/3! ...] 1 0 0 ... 1

Therefore, the limit is:

limx→0 (e^x - 1) / x 1

This confirms that the limit is indeed 1.

Conclusion

In summary, we have shown that the limit of (e^x - 1) / x as x approaches 0 is 1 using three different methods: the derivative definition, L'H?pital's rule, and Taylor series expansion. Each method provides a unique insight into the behavior of the function as x gets infinitesimally small.