Problem 48 - Calculus Early Transcendentals

Question

Without using any trigonometric identities, find $$ \lim _{x \rightarrow 0} \frac{\tan (x+y)-\tan y}{x} $$ [Hint: Relate the given limit to the definition of the derivative of an appropriate function of y .]

Step-by-Step Solution

Answer

The limit is \( \sec^2(y) \).

Step 1: Understanding the Problem

The problem asks us to find the limit \( \lim _{x \rightarrow 0} \frac{\tan(x+y)-\tan y}{x} \). Observing the structure, it resembles the definition of the derivative of a function at a point.

Step 2: Recognize a Derivative Form

The given expression can be related to the definition of a derivative: \( f'(y) = \lim_{x \to 0} \frac{f(y + x) - f(y)}{x} \). Comparing with our limit, set \( f(y) = \tan(y) \). Therefore, the problem reduces to finding \( \frac{d}{dy}[\tan(y)] \) at the point \( y \).

Step 3: Find the Derivative of \( \tan(y) \)

The derivative of \( \tan(y) \) with respect to \( y \) is known to be \( \sec^2(y) \), which can be derived using the quotient rule or chain rule, by considering \( \tan(y) = \frac{\sin(y)}{\cos(y)} \). Hence, \( f'(y) = \sec^2(y) \).

Step 4: Evaluate the Limit

Since \( \lim_{x \to 0} \frac{\tan(x+y) - \tan(y)}{x} \) is essentially \( \frac{d}{dy}[\tan(y)] \), the limit is equal to \( \sec^2(y) \).

Key Concepts

Limit

Limits are a fundamental concept in calculus. They help us understand how a function behaves as its input approaches a certain value. When you think of a limit, imagine the point a function heads towards as you get closer and closer to a specific input.
For example, consider the expression \[ \lim_{x \rightarrow 0} \frac{\tan(x+y) - \tan(y)}{x} \]. This is practically asking: what happens to this fraction when \( x \) gets very close to 0? Notice how this looks a lot like the derivative definition, which deals with finding the slope or rate of change of a function at a particular point.

• The fraction essentially suggests a tiny change in the function \( \tan \) with a tiny change in \( x \).
• The value of the limit gives us how the function \( \tan(x+y) \) behaves near \( x = 0 \).

Understanding limits is key when it comes to grasping the concept of derivatives, as they shape how we describe changes in functions.

Tangent Function

The tangent function, \( \tan(y) \), is one of the fundamental trigonometric functions. It describes a specific relationship between angles and sides in a right triangle. In a unit circle, \( \tan(y) \) is the ratio of the sine and cosine of an angle \( y \), or mathematically, \( \tan(y) = \frac{\sin(y)}{\cos(y)} \).
What makes the tangent function particularly interesting in calculus is how it changes, or its derivative. This derivative represents the rate at which \( \tan(y) \) changes as \( y \) increases. When calculating the derivative of \( \tan(y) \), it turns out to be \( \sec^2(y) \), using rules like chain rule or quotient rule.

• This relationship is crucial especially when we deal with problems involving trigonometric limits.
• Finding the derivative of \( \tan(x+y) - \tan(y) \) effectively helps us find the limit in our original problem.

This derivative highlights the importance of the tangent function in a wide range of calculus problems.

Chain Rule

The chain rule is a powerful tool in calculus for finding the derivative of complex functions. If you have a function of a function, this rule helps you differentiate it step by step.
Let's see how this works: consider a function \( g(y)\) inside another function, \( f(g(y)) \). The chain rule tells us that the derivative of this composite function is \( f'(g(y)) \cdot g'(y) \).

• This formula helps break down complicated expressions into simpler parts.
• It's especially useful for trigonometric functions like \( \tan(y) \), which itself can be seen as a composition of more basic functions.

By applying the chain rule, we can effectively determine how each part of a complex function contributes to the overall rate of change, making problems dealing with derivatives more manageable and clear.