Evaluating Limits Involving Exponential and Trigonometric Functions

Understanding how to evaluate limits involving exponential and trigonometric functions is a fundamental skill in calculus. In this article, we will focus on evaluating a specific limit and discuss the methodologies and steps involved. We will explore different transformations and series expansions to simplify the evaluation process.

The Limit to Evaluate

Consider the limit:

$$lim_{x to 0} frac{e^x e^{-x} - 2}{1 - cos x}$$

Step-by-Step Evaluation

Let's break down the evaluation process into manageable steps. The key is to simplify the expression inside the limit by using known identities and series expansions.

Initial Transformation

First, we rewrite the numerator using the identity for the hyperbolic sine function:

$$e^x e^{-x} - 2 2 sinh x$$

Thus, the limit becomes:

$$lim_{x to 0} frac{2 sinh x}{1 - cos x}$$

Using the Hyperbolic Identity

We know that:

$$sinh x frac{e^x - e^{-x}}{2}$$

Therefore, the limit can be rewritten as:

$$lim_{x to 0} frac{2 left(frac{e^x - e^{-x}}{2}right)}{1 - cos x} lim_{x to 0} frac{e^x - e^{-x}}{1 - cos x}$$

Further Simplification

Next, we use the identity for the hyperbolic sine squared:

$$sinh^2 x frac{e^{2x} - 2 e^{-2x}}{4}$$

Thus, the limit becomes:

$$lim_{x to 0} frac{2 sinh^2 frac{x}{2}}{1 - cos x}$$

Using the Sine Series Expansion

Now, we use the series expansion for sine and cosine:

$$sin x approx x - frac{x^3}{3!} frac{x^5}{5!} - cdots$$

$$cos x approx 1 - frac{x^2}{2!} frac{x^4}{4!} - cdots$$

Substituting these into the limit, we get:

$$lim_{x to 0} frac{2 left(frac{(frac{x}{2})^2}{2!} - frac{(frac{x}{2})^4}{4!} cdotsright)}{left(1 - frac{x^2}{2!} cdotsright) - left(1 - frac{x^2}{2} cdotsright)} lim_{x to 0} frac{2 left(frac{x^2}{8} - frac{x^4}{384} cdotsright)}{frac{x^2}{2}} 2$$

Pointwise Substitution

Alternatively, we can use a pointwise substitution:

Let (z frac{x}{2})

When (x to 0), (z to 0)

$$lim_{x to 0} frac{e^x e^{-x} - 2}{1 - cos x} lim_{z to 0} frac{4 sinh^2 z}{sin^2 z} 2 lim_{z to 0} left(frac{sinh z}{z}right)^2 2$$

Final Conclusion

The limit evaluates to 2. This result is obtained using a combination of series expansions and direct substitution techniques.

Conclusion

Evaluating limits involving exponential and trigonometric functions can be complex, but with the right transformations and series expansions, it becomes more manageable. By understanding these techniques, you can solve similar problems efficiently and accurately.