Understanding the Limit of lim_{x→0} x / (x * tan(x))

When faced with evaluating the limit of the function limx→0 x / (x * tan(x)), we encounter a common scenario in calculus where direct substitution leads to an indeterminate form (0/0). However, this form is not insurmountable, and we can apply L'H?pital's Rule to resolve it.

L'H?pital's Rule and Indeterminate Forms

When evaluating a limit of a function that results in an indeterminate form (such as 0/0 or ∞/∞), L'H?pital's Rule provides a means to find the limit. This rule states that if the limit of the ratio of two functions as x approaches a certain value is of an indeterminate form, then the limit can often be found by taking the derivative of the numerator and the denominator and then evaluating the limit of this new ratio.

Applying L'H?pital's Rule to lim_{x→0} x / (x * tan(x))

Let's break down the process step by step:

Firstly, we observe the original limit:

[lim_{xto 0} frac{x}{x tan(x)}]

Direct substitution at x 0 gives us 0/0, which is an indeterminate form. This is where L'H?pital's Rule comes into play. We differentiate the numerator and the denominator separately and evaluate the limit again:

[lim_{xto 0} frac{frac{d}{dx}(x)}{frac{d}{dx}(x tan(x))}]

Calculating the derivatives, we get:

[lim_{xto 0} frac{1}{1 tan^2(x)} frac{1}{1 frac{sin^2(x)}{cos^2(x)}}]

Recall the trigonometric identity (tan(x) frac{sin(x)}{cos(x)}). Therefore, we can rewrite the expression further:

[lim_{xto 0} frac{1}{frac{1}{cos^2(x)}} frac{cos^2(x)}{1} cos^2(x)]

Evaluating this as x approaches 0, we get:

[cos^2(0) 1^2 1]

Thus, the limit (lim_{xto 0} frac{x}{x tan(x)} frac{1}{1 1} frac{1}{2}).

Step-by-Step Solution

Let's go through the detailed steps again for clarity:

Identify the form as 0/0 when x approaches 0. Apply L'H?pital's Rule: Numerator: (frac{d}{dx}(x) 1) Denominator: (frac{d}{dx}(x tan(x)) x' tan(x) x tan'(x) tan(x) x sec^2(x)) Simplify the new limit: (lim_{xto 0} frac{1}{1 tan^2(x)}) Recognize the trigonometric identity (1 tan^2(x) frac{1}{cos^2(x)}) Thus, (lim_{xto 0} frac{cos^2(x)}{1} cos^2(0) 1) Conclude that (lim_{xto 0} frac{x}{x tan(x)} frac{1}{1 1} frac{1}{2}).

Conclusion

The limit of (frac{x}{x tan(x)}) as x approaches 0 is (frac{1}{2}). This process showcases the power of L'H?pital's Rule in simplifying and resolving indeterminate forms in trigonometric limits.

For more calculus and limit problems, visit our resources page.