Question

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0}(\sin 5 x) / 3 x $$

Short answer

The limit is \( \frac{5}{3} \).

Step-by-step solution

Understand the Limit

We are asked to find \( \lim_{x \rightarrow 0} \frac{\sin 5x}{3x} \). This involves understanding how the values behave as \( x \) approaches zero.

Recall the Squeeze Theorem

To find the limit, we can use the known limit \( \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 \). This will help us simplify the expression because of its similarity to this standard limit form.

Factor and Simplify

We observe that \( \frac{\sin 5x}{3x} = \frac{5}{3} \cdot \frac{\sin 5x}{5x} \). This allows us to separate the constants and focus on the limit involving the sine function.

Substitute Known Limit Value

We substitute \( \lim_{x \rightarrow 0} \frac{\sin 5x}{5x} = 1 \) based on the standard limit form earlier mentioned. Therefore, the expression becomes \( \frac{5}{3} \times 1 \).

Calculate Final Result

Multiply the constant to get the final result for the limit. Therefore, the value of the limit is \( \frac{5}{3} \).

Key concepts

Squeeze Theorem

The Squeeze Theorem, also known as the Sandwich Theorem, is a powerful tool in calculus that helps us evaluate limits of functions that are difficult to assess directly. This theorem states that if you have three functions \( f(x) \), \( g(x) \), and \( h(x) \) such that \( f(x) \leq g(x) \leq h(x) \) for all \( x \) near a point \( a \) (not necessarily at \( a \)), and if \( \lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L \), then \( \lim_{x \to a} g(x) = L \).

This means if a function is "squeezed" between two other functions that both approach the same limit, then the middle function also approaches that limit. This theorem is particularly useful when dealing with trigonometric and oscillating functions, where visualization can be challenging.

To use this in a trigonometric context, often trigonometric inequalities are established based on well-defined bounds, as seen with the limit \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). This gives a pathway to use algebraic manipulations and known boundary function limits to establish complex limit solutions.

standard limit form

In calculus, the standard limit form \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) is known as a fundamental limit in trigonometry. This limit is useful because it tells us how the sine function behaves very close to zero.

The intuitive understanding behind \( \frac{\sin x}{x} \) approaching 1 as \( x \to 0 \) comes from the geometrical fact that the sine of an angle in radians is approximately equal to the angle itself when the angle is very small. This is because, in a unit circle, the arc length (which equals the angle in radians) is roughly equivalent to its vertical height (\( \sin x \)), especially as the angle diminishes.

Because of its fundamental nature, this standard limit form sets a guideline for solving many other trigonometrically-based limits, especially when combined with other mathematical principles such as the Squeeze Theorem or algebraic simplification.

trigonometric limits

Trigonometric limits refer to the evaluation of limits involving trigonometric functions as the input approaches a certain value. These limits are crucial for understanding the behavior of functions, especially in oscillating scenarios typical of trigonometric functions.

Some common trigonometric limits that are often encountered include:

• \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \)
• \( \lim_{x \to 0} \frac{\tan x}{x} = 1 \)
• \( \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \)

These limits often serve as the cornerstones for more complex problems.

Many trigonometric limit problems, like the one we are solving with \( \lim _{x \rightarrow 0} (\sin 5 x) / 3 x \), require creative rearrangements and sometimes using properties of trig identities to simplify or match them to these known limit forms. Recognizing these patterns simplifies and speeds up the calculative process, underlining the importance of familiarity with these fundamental limits.