Problem 68 - Calculus of a Single Variable

Question

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 0} \frac{\tan ^{2} x}{x} $$

Step-by-Step Solution

Answer

The limit of the function \(\frac{\tan^{2} x}{x}\) as x approaches 0 is 0.

Step 1: Rewrite the function

Instead of \(\frac{\tan ^{2} x}{x}\), the function can be written as \(\frac{\tan x}{\sqrt{x}}\times\frac{\tan x}{\sqrt{x}}.\) This form can be derived from the property of square where \(a^2 = a \times a\). This step is important because it allows breaking down the function into simpler terms, where known limits could be applied.

Step 2: Apply Limit Properties

Limit of a product is the product of the limits, if both limits exist. Therefore \(\lim _{x \rightarrow 0} \frac{\tan x}{\sqrt{x}}\times\frac{\tan x}{\sqrt{x}} = \lim _{x \rightarrow 0} (\frac{\tan x}{\sqrt{x}}) ^2\). It is important to note that now we need to find the limit of \(\frac{\tan x}{\sqrt{x}}\) as this limit squared will be our answer.

Step 3: Apply L'Hopital's Rule

The L'Hopital's Rule states that the limit of the ratio of two functions as x approaches a certain value is equal to the limit of the ratio of their derivatives. In particular, it can be applied when the original limit is in indeterminate form (0/0 or ∞/∞). The derivative of \(\tan x\) is \(\sec^2 x\) and the derivative of \(\sqrt{x}\) is \(\frac{1}{2\sqrt{x}}\). Applying L'Hopital's rule, the limit becomes \(\lim _{x \rightarrow 0} \frac{\sec^2 x}{\frac{1}{2\sqrt{x}}}\), which can be simplified to \(\lim _{x \rightarrow 0} 2\sec^2 x \sqrt{x}\).

Step 4: Evaluate the Limit of the Simplified Expression

Once again using limit properties, we can separate this to two limits, one for \(2\sec^2 x\), which equals 2, and the other one for \(\sqrt{x}\), which is 0. Therefore, the limit of the whole expression is 0.

Step 5: Solution

The final step is to square the result obtained in Step 4. So, \(0^2 = 0\) is our solution.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool used in calculus for finding limits of indeterminate forms. Indeterminate forms are cases where a function evaluates to expressions like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). L'Hôpital's Rule provides a way to handle these confusing calculations by allowing you to take the derivatives of the numerator and the denominator to find the limit.

To apply L'Hôpital's Rule, follow these steps:

• First, confirm that the limit of the function is in an indeterminate form, like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
• Next, compute the derivatives of the numerator and the denominator.
• Then, take the limit of the new fraction formed by these derivatives.

It’s important to remember that L'Hôpital's Rule can only be applied if the function remains indeterminate after differentiating. If not, you should consider other methods, like algebraic manipulation, to solve the limit.

Indeterminate Form

An indeterminate form arises when you're trying to calculate the limit of a function, but the function behaves in a way that isn’t immediately clear. The most common indeterminate forms are \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\).

Indeterminate forms occur because, mathematically, you cannot divide by zero or have infinite values define a clear outcome. Instead of concluding that the limit does not exist or taking a guess, advanced calculus techniques like L'Hôpital's Rule are used to understand the behavior of the function near the troublesome point.

For instance, consider the function \( f(x) = \frac{\tan^2 x}{x} \) as \( x \to 0 \). At first glance, replacing \( x \) with 0 gives us \( \frac{0}{0} \), a classic indeterminate form. To proceed, you can apply L'Hôpital's Rule, which is suitable for resolving such forms. Remember that sometimes solving these requires simplifying, factoring, or rationalizing the function before applying more advanced techniques.

Trigonometric Limits

Trigonometric limits are an important part of calculus, especially when dealing with periodic functions like sine, cosine, and tangent. Understanding these limits allows for the calculation of more complex expressions that involve trigonometric functions.

One useful limit to recall is\[\lim_{x \to 0} \frac{\sin x}{x} = 1.\]This limit helps in simplifying expressions involving small angles.

When evaluating a limit like \( \lim_{x \to 0} \frac{\tan^2 x}{x} \), trigonometric identities can be helpful. For example, you can express \( \tan x \) in terms of \( \sin x \) and \( \cos x \), which helps in applying known limits. Moreover, the derivative properties of trigonometric functions are often used alongside L'Hôpital's Rule to simplify these evaluations further.

Using trigonometric limits effectively often requires a blend of identities and calculus techniques. As you practice, identifying which strategies work best in different scenarios will become clearer.