Problem 65 - Precalculus with Limits

Question

In Exercises 49-68, find the limit by direct substitution. $$ \lim_{x \to \pi}\ \textrm{sin}\ 2x$$

Step-by-Step Solution

Answer

The limit of \(\textrm{sin}\ 2x\) as \(x \to \pi\) is 0.

Step 1: Identify the function

Here, the function is the sine function or \(\textrm{sin}\ 2x\). And we need to find its limit as \(x\to\pi\).

Step 2: Apply the limit

Replacing \(x\) with \(\pi\) in the function, we get \(\textrm{sin}\ 2\pi\).

Step 3: Evaluate the sine function

The sine of any multiple of \(\pi\) is zero. Therefore, \(\textrm{sin}\ 2\pi = 0\)

Key Concepts

Direct substitution method

The direct substitution method is a straightforward approach to finding limits in calculus. When you use this method, you directly substitute the value approaching in the limit into the function. For example, in the limit \[ \lim_{x \to \pi} \sin 2x \] we replace \(x\) with \(\pi\). This substitution gives us \(\sin 2\pi\). This method is only valid when the function is continuous at the point where the limit is being evaluated.

• If a function is continuous at a certain point, substituting the value of \(x\) directly yields the limit.
• If the function is not defined or not continuous at the point, other methods like L'Hôpital's rule or algebraic manipulation might be required.

The advantage of the direct substitution method is its simplicity, making it a preferred initial approach when dealing with limits.

Trigonometric limits

Trigonometric limits often involve functions like sine, cosine, and tangent. Understanding the properties and behaviors of these functions is crucial in evaluating limits involving them. The limit \[ \lim_{x \to \pi} \sin 2x \] is a classic case of applying knowledge about the sine function's periodicity and properties. The sine and cosine functions are continuous for all values of \(x\), meaning the direct substitution method can typically be used.

• Trigonometric functions like sine have known values at key angles (\(0\), \(\pi/2\), \(\pi\), etc). This knowledge simplifies limit evaluations.
• These functions are periodic, so they repeat their values in predictable intervals, which helps in anticipating the behavior of their limits.

Understanding trigonometric limits requires familiarity with the unit circle and key angle values.

Sine function

The sine function, written as \(\sin x\), is fundamental in both algebra and calculus. It represents the y-coordinate of a point on the unit circle as the angle \(x\) varies. Here are a few key features of the sine function:

• The sine function is periodic with a period of \(2\pi\), meaning that \(\sin(x + 2\pi k) = \sin x\) for any integer \(k\).
• Key values: \(\sin 0 = 0\), \(\sin \pi = 0\), \(\sin \frac{\pi}{2} = 1\), and \(\sin \frac{3\pi}{2} = -1\).

When evaluating \(\lim_{x \to \pi} \sin 2x \), we found that \(\sin 2\pi\) equals 0, as the sine of any integer multiple of \(\pi\) is zero due to its oscillatory nature. This property makes solving limits involving sine more straightforward when substituting known angle values.