Problem 53 - University Calculus: Early Transcendentals

Question

Find the limits. $$\lim _{t \rightarrow 0} \tan \left(1-\frac{\sin t}{t}\right)$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Analyze the Expression

We need to determine the limit as \( t \rightarrow 0 \) of the expression \( \tan(1 - \frac{\sin t}{t}) \). Currently, there is no immediate substitution we can use to find this limit due to the form of \( \frac{\sin t}{t} \).

Step 2: Apply Limit Properties

To find the limit, recall that \( \lim_{t \to 0} \frac{\sin t}{t} = 1 \). This simplifies \( 1 - \frac{\sin t}{t} \) as \( t \) approaches 0 becomes \( 1 - 1 = 0 \).

Step 3: Substitute Limit Results

Substitute the result from Step 2 into the original expression so that it becomes \( \tan(0) \). The tangent function \( \tan(x) \) approaches 0 as \( x \rightarrow 0 \).

Step 4: Conclusion on Limit Value

Therefore, the limit \( \lim_{t \to 0} \tan \left(1- \frac{\sin t}{t} \right) = 0 \).

Key Concepts

Tangent Function

The tangent function, often abbreviated as \( \tan(x) \), is a fundamental trigonometric function. Its behavior is cyclical and it is important in describing various periodic phenomena. At its core, the tangent function can be described as the ratio of the sine function to the cosine function:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

This function is particularly interesting near angles where \( \cos(x) \) approaches zero, as it tends to infinity around those points. However, around zero, the tangent function behaves nicely and approaches zero itself. This is essential for using the tangent function in limits, as it simplifies many trigonometric expressions when evaluating limits. When analyzing limits involving tangent, we often replace complex trigonometric expressions with their simpler equivalent forms near zero.

Sine Function

The sine function, denoted as \( \sin(x) \), is another critical trigonometric function. It defines the y-coordinate of a point on the unit circle as it sweeps through angles. Key properties of the sine function include:

• Range: The values of \( \sin(x) \) are bounded between -1 and 1.
• Periodicity: \( \sin(x) \) is periodic with a period of \( 2\pi \).
• Symmetry: \( \sin(-x) = -\sin(x) \), signifying its odd symmetry.

When assessing limits, a special case appears frequently: \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). This identity is crucial for solving trigonometric limits and often simplifies complex expressions.

Limit Properties

Limits are foundational in calculus. They describe the behavior of a function as it approaches a certain point. Some key properties of limits that aid in simplifying complex expressions include:

• Substitution Property: If a function is continuous at a point, you can simply substitute the point into the function to find the limit.
• Operations on Limits: You can add, subtract, multiply, and divide limits, provided the resultant operations are defined (e.g., you can't divide by zero).
• Sandwich Theorem (Squeeze Theorem): It is useful when you can bound a difficult-to-evaluate function between two easier ones that converge to the same limit.

By applying these properties, you can break down daunting expressions into simpler forms, making it easier to evaluate limits precisely.

Trigonometric Limits

Trigonometric limits are limits that involve trigonometric functions like sine, cosine, and tangent. These are crucial in calculus for simplifying expressions and finding precise limit values.

• A pivotal standard limit is \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \).
• Another is \( \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \), which often helps in solving more complex limits.

When tackling trigonometric limits, one strategy is to rewrite or approximate the trigonometric functions using these special limits or use identities to reduce complexity. Mastering these limits is essential for solving a wide range of calculus problems involving both basic trigonometric functions and their composite functions, as is often the case in calculus.