Problem 23 - Calculus for Biology and Medicine

Question

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} \frac{\sin x}{x} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Recognize the Type of Limit

We are dealing with the limit \( \lim_{x \to 0} \frac{\sin x}{x} \). This is a well-known trigonometric limit often encountered in calculus.

Step 2: Use Trigonometric Limit Identity

The identity \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) is a standard result in calculus. This can be derived using the squeeze theorem, but it's important to remember it for problems involving limits.

Step 3: Verification with L'Hôpital's Rule (Optional)

If you are unfamiliar with the trigonometric limit, you could apply l'Hôpital's Rule since this limit initially appears as \( \frac{0}{0} \) as \( x \to 0 \), an indeterminate form. Differentiate the numerator and the denominator: \( \frac{d}{dx}(\sin x) = \cos x \) and \( \frac{d}{dx}(x) = 1 \). So the limit becomes \( \lim_{x \to 0} \frac{\cos x}{1} = \cos(0) = 1 \).

Key Concepts

Limits

A limit in calculus is a way to determine the value that a function or sequence "approaches" as the input (or index) approaches some value. When we say "the limit of a function \( f(x) \) as \( x \) approaches \( c \) is \( L \)," we mean that as \( x \) gets closer and closer to \( c \), \( f(x) \) gets closer and closer to \( L \). This is written as:\[\lim_{x \to c} f(x) = L\]Understanding limits is fundamental in calculus as they form the basis for concepts like derivatives and integrals.

• **Approach:** A limit looks at what value a function is approaching, not necessarily the value it takes.
• **Existence:** A limit exists if the function approaches the same value from both sides (left and right) as \( x \) approaches \( c \).
• **Indeterminate Forms:** Some limits may initially appear uncertain, such as \( \frac{0}{0} \). These require special techniques or theorems, like L'Hôpital's Rule, to resolve.

Limits help deal with continuous change scenarios, determining values like slopes of tangent lines or the area under curves, underpinning much of what calculus encompasses.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool for evaluating limits involving indeterminate forms, particularly \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \). When a limit problem results in one of these forms, you can use L'Hôpital's Rule by taking the derivative of the numerator and denominator separately. The rule states:\[\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\]Provided the following:

• Both \( f(x) \) and \( g(x) \) are differentiable near \( c \).
• \( g'(x) eq 0 \) near \( c \).

To apply L'Hôpital's Rule, ensure that you differentiate correctly and continue applying it if the result is still an indeterminate form. Be cautious, however, as not all limits can be resolved by this method, especially if the derivatives do not lead to a clearer form. But for many \( \frac{0}{0} \) cases, it's a great tool to transition from complex limits to simpler solutions.

Trigonometric Limits

Trigonometric limits often appear in calculus, particularly when dealing with angle-based problems or periodic phenomena. An essential trigonometric limit to remember is:\[\lim_{x \to 0} \frac{\sin x}{x} = 1\]This limit is fundamental due to its applications in both calculus and analytical geometry. Here's why you should commit it to memory:

• **Common Use:** You'll encounter this limit extensively across calculus problems; it allows for simplifying expressions directly.
• **Proofs Involve:** While often used as a formula, its derivation can be shown using the squeeze theorem or unit circle analysis.
• **Foundation for Series:** It's crucial when developing series expansions for trigonometric functions.

Trigonometric limits like \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) form the building blocks for understanding how trigonometric functions behave infinitely close to specific points. They help in evaluating limits quickly or proving continuity of functions at those points.