Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each limit. $$ \lim _{t \rightarrow 0} \frac{\tan ^{2} 3 t}{2 t} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the form of the limit

Begin by identifying the form of the limit. As \( t \to 0 \), \( \tan^2(3t) \) and \( 2t \) both approach 0, indicating an indeterminate form \( \frac{0}{0} \).

Step 2: Apply Trigonometric Identity

Recall the trigonometric identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Thus, \( \tan^2(3t) = \left(\frac{\sin(3t)}{\cos(3t)}\right)^2 = \frac{\sin^2(3t)}{\cos^2(3t)} \). The limit becomes \( \lim_{t \to 0} \frac{\sin^2(3t)}{2t \cos^2(3t)} \).

Step 3: Use Limit Property for Sine

Apply the limit property \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \). We transform \( \sin^2(3t) \) into \( 9t^2 \left(\frac{\sin(3t)}{3t}\right)^2 \). Substitute to get \( \lim_{t \to 0} \frac{9t^2 (\frac{\sin(3t)}{3t})^2}{2t \cos^2(3t)} \).

Step 4: Simplify and Evaluate the Limit

Simplify the expression: \( \frac{9t^2}{2t \cos^2(3t)} \times (\frac{\sin(3t)}{3t})^2 = \frac{9t}{2 \cos^2(3t)} \times (1)^2 \). Substitute \( t = 0 \) to get \( \frac{9 \times 0}{2 \times 1} = 0 \).

Step 5: Conclusion

Concluding, after simplifications using trigonometric identities and limit properties, the given limit evaluates to 0.

Key Concepts

Trigonometric Limits

Trigonometric limits often involve expressions where trigonometric functions approach a specific point, causing them to tend towards infinity or form indeterminate expressions. When dealing with trigonometric limits, such as the given exercise,\[ \lim_{t \to 0} \frac{\tan^2(3t)}{2t}, \]we encounter trigonometric functions like \(\tan(x)\). These functions can behave unpredictably as their argument approaches certain critical values.To handle these scenarios, we exploit trigonometric identities.

• Recalling the identity \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), which helps reform expressions.
• This assists in transforming the limit into a more manageable form, such as dealing with \(\sin(x)\) and \(\cos(x)\).

By transforming \(\tan^2(3t)\) into \(\frac{\sin^2(3t)}{\cos^2(3t)}\), the expression becomes easier to manipulate and can be simplified using known limit rules.

Indeterminate Forms

Indeterminate forms occur in calculus when standard arithmetic operations do not directly resolve to a concrete value as a variable approaches a limit. These commonly include forms like \(\frac{0}{0}\), \(\frac{\infty}{fty}\), or \(0 \times \infty\).In our exercise,\[ \lim_{t \to 0} \frac{\tan^2(3t)}{2t}, \]both the numerator \(\tan^2(3t)\) and the denominator \(2t\) approach zero. This means the expression becomes a \(\frac{0}{0}\) indeterminate form when \(t\) tends to zero.To address this, we need to apply techniques that transform the expression into a determinate form.

• Utilizing trigonometric identities, such as those involving sine and cosine, helps to eliminate indeterminacy.
• Applying specific limit properties can simplify the expression further, often converting it into recognizable standard limit problems.

Effectively handling indeterminate forms requires a combination of algebraic manipulation and applying limit theorems.

Limit Evaluation Techniques

Limit evaluation techniques are essential when directly substituting values into an expression results in indeterminate forms like \(\frac{0}{0}\). These techniques help us find meaningful limits.In this example:\[ \lim_{t \to 0} \frac{\tan^2(3t)}{2t}, \]we apply several methods:

• **Trigonometric Identities**: By converting \(\tan^2(3t)\) into \(\frac{\sin^2(3t)}{\cos^2(3t)}\), we manipulate the expression into a simpler form.


• **Using \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\)**: This key limit property helps when reasoning about expressions involving \(\sin(3t)\).


• **Algebraic Simplification**: Breaking down the complex limit into smaller, more manageable pieces, makes use of the relation \((\frac{\sin(3t)}{3t})^2\) which resolves to \(1^2 = 1\).

Applying these techniques step-by-step leads us to the result of 0, showcasing the process from indeterminate form to a meaningful limit.