Understanding the Limit of tan2x / (1 - cosx) as x Approaches 0

In this article, we will explore the limit of (frac{tan^2 x}{1 - cos x}) as (x) approaches 0. We will use two different methods: Taylor series expansion and L'H?pital's rule. This exploration will provide a deep understanding of how to handle indeterminate forms in limit problems.

Method 1: Taylor Series Expansion

The Taylor series expansion is a powerful tool for approximating functions. We will expand (tan x) and (cos x) around (x 0) to find the limit. The expansions are as follows:

(tan x x frac{x^3}{3} O(x^5))

(cos x 1 - frac{x^2}{2} O(x^4))

Step 1: Expand tan2x

(tan^2 x left(x frac{x^3}{3} O(x^5)right)^2 x^2 frac{2x^4}{3} O(x^6))

Step 2: Expand 1 - cosx

(1 - cos x frac{x^2}{2} O(x^4))

Step 3: Substitute into the limit

(lim_{x rightarrow 0} frac{tan^2 x}{1 - cos x} lim_{x rightarrow 0} frac{x^2 frac{2x^4}{3} O(x^6)}{frac{x^2}{2} O(x^4)})

Step 4: Simplify the expression

Dividing the numerator and denominator by (x^2):

(frac{1 frac{2x^2}{3} O(x^4)}{frac{1}{2} O(x^2)})

As (x) approaches 0, the (O(x^2)) terms vanish, leaving:

(lim_{x rightarrow 0} frac{1 O(x^2)}{frac{1}{2} O(x^2)} 2)

Method 2: L'H?pital's Rule

Since both the numerator and denominator approach 0 as (x) approaches 0, L'H?pital's rule can be applied.

Step 1: Differentiate the numerator and denominator

Numerator: (frac{d}{dx} tan^2 x 2tan x sec^2 x)

Denominator: (frac{d}{dx} 1 - cos x sin x)

Step 2: Apply L'H?pital's rule

(lim_{x rightarrow 0} frac{tan^2 x}{1 - cos x} lim_{x rightarrow 0} frac{2tan x sec^2 x}{sin x})

Step 3: Evaluate as x approaches 0

Using the small angle approximations: (tan x sim x), (sec^2 x sim 1), and (sin x sim x) as (x) approaches 0:

(lim_{x rightarrow 0} frac{2tan x sec^2 x}{sin x} lim_{x rightarrow 0} frac{2x cdot 1}{x} 2)

Conclusion

Both methods yield the same result. Therefore, the limit is:

(lim_{x rightarrow 0} frac{tan^2 x}{1 - cos x} 2)

Additional Insights

For further exploration, consider the following related limits and their implications in calculus:

Limit of tan2x as x approaches 0

(lim_{x rightarrow 0} tan^2 x 0)

Limit of 1 - cosx as x approaches 0

(lim_{x rightarrow 0} 1 - cos x 0)

Understanding these limits can help in grasping the behavior of trigonometric functions around (x 0) and their applications in more complex problems.