Understanding the Trigonometric Limit Problem

We are asked to find the value of the limit \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x{}^\circ }{\tan 3x{}^\circ }\). This involves evaluating a limit of a ratio of trigonometric functions as \(x\) approaches 0. A key aspect of this problem is that the angles are given in degrees (\(x^\circ\) and \(3x^\circ\)), while the standard trigonometric limit formulas, such as \(\underset{\theta\to 0}{\mathop{\lim }}\,\frac{\sin \theta}{\theta} = 1\) and \(\underset{\theta\to 0}{\mathop{\lim }}\,\frac{\tan \theta}{\theta} = 1\), require the angle \(\theta\) to be in radians.

Converting Degrees to Radians for Limit Evaluation

To use the standard limit formulas, we must convert the angles from degrees to radians. The conversion factor is \(\pi\) radians per 180 degrees. So, to convert an angle from degrees to radians, we multiply the degree measure by \(\frac{\pi}{180}\).

• For \(x^\circ\): \(x^\circ = x \times \frac{\pi}{180}\) radians.
• For \(3x^\circ\): \(3x^\circ = 3x \times \frac{\pi}{180}\) radians.

Now, we can rewrite the given limit expression with angles in radians:

\(\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \left(x \times \frac{\pi}{180}\right)}{\tan \left(3x \times \frac{\pi}{180}\right)}\)

Let \(\theta_1 = x \times \frac{\pi}{180}\) and \(\theta_2 = 3x \times \frac{\pi}{180}\). As \(x \to 0\), both \(\theta_1 \to 0\) and \(\theta_2 \to 0\).

Step-by-Step Calculation of the Limit

We can now use the standard limits \(\underset{\theta\to 0}{\mathop{\lim }}\,\frac{\sin \theta}{\theta} = 1\) and \(\underset{\theta\to 0}{\mathop{\lim }}\,\frac{\tan \theta}{\theta} = 1\). To do this, we will multiply and divide the numerator and denominator by the arguments of the sine and tangent functions, respectively.

The limit is:

\(\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \left(\frac{\pi x}{180}\right)}{\tan \left(\frac{3\pi x}{180}\right)}\)

Multiply and divide by the arguments:

\(\underset{x\to 0}{\mathop{\lim }}\,\frac{\frac{\sin \left(\frac{\pi x}{180}\right)}{\frac{\pi x}{180}} \times \left(\frac{\pi x}{180}\right)}{\frac{\tan \left(\frac{3\pi x}{180}\right)}{\frac{3\pi x}{180}} \times \left(\frac{3\pi x}{180}\right)}\)

As \(x \to 0\), we know that:

• \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin \left(\frac{\pi x}{180}\right)}{\frac{\pi x}{180}} = 1\) (using \(\theta = \frac{\pi x}{180}\))
• \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan \left(\frac{3\pi x}{180}\right)}{\frac{3\pi x}{180}} = 1\) (using \(\theta = \frac{3\pi x}{180}\))

So, the limit becomes:

\(\underset{x\to 0}{\mathop{\lim }}\,\frac{1 \times \left(\frac{\pi x}{180}\right)}{1 \times \left(\frac{3\pi x}{180}\right)}\)

Now, we simplify the remaining expression:

\(\underset{x\to 0}{\mathop{\lim }}\,\frac{\frac{\pi x}{180}}{\frac{3\pi x}{180}}\)

Since \(x \to 0\), \(x\) is close to but not equal to 0, so \(\frac{\pi x}{180}\) is non-zero. We can cancel the common term \(\frac{\pi x}{180}\) from the numerator and the denominator:

\(\frac{1}{3}\)

Thus, the value of the limit is \(\frac{1}{3}\).

Revision Table: Key Limit Concepts

Additional Information: Understanding Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as the input approaches a particular value. In this problem, we evaluated a limit involving trigonometric functions. It's crucial to remember that many calculus formulas and identities for trigonometric functions are based on the assumption that angles are measured in radians. Failing to convert from degrees to radians is a common mistake when evaluating such limits.

The technique used here, multiplying and dividing by a term to match the standard limit form, is a very common and useful method for evaluating trigonometric limits. Other techniques for evaluating limits include direct substitution (if the function is continuous at the point), factorization, rationalization, and L'Hopital's Rule (though the latter is typically introduced after derivatives).