Problem 66 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The limit of the function as x approaches pi is 0.

Step 1: Understand the concept

Getting familiar with the trigonometric function 'tan' and what the limit of 'tan x' as x approaches some values will be. Particularly, the tangent function is undefined at \( \pi/2 + k \pi \) for any integer value of k and hence it is necessary to make sure that the x-value the limit approaches isn't such a value.

Step 2: Perform direct substitution

In the given limit \( \lim_{x \to \pi}\ \textrm{tan}\ x \), plug in \( x = \pi \) into the function, which gives \( \textrm{tan}\ \pi \). The tangent of \(\pi\) radians is 0 in the unit circle.

Step 3: Conclusion

Therefore, the limit of the function \( \textrm{tan} x \) as x approaches \( \pi \) is 0.

Key Concepts

Direct Substitution Method

The direct substitution method is a foundational technique in calculus for finding limits of functions. It is precisely as simple as it sounds: to find the limit as the variable approaches a certain value, you substitute that value into the function.

When applying the direct substitution method, it’s crucial to check if the function is continuous at that point. If the function is defined and continuous at the value you're approaching, direct substitution typically gives you the correct limit. However, if the function is not continuous, other methods, like factoring or rationalizing, may be required to evaluate the limit.

Trigonometric Functions

Trigonometric functions are fundamental in calculus, especially when dealing with angles and periodic phenomena. The main trigonometric functions are sine, cosine, and tangent, each with specific properties and graphs.

The sine and cosine functions take an angle as input and return the y-coordinate and x-coordinate respectively, of a point on the unit circle. The tangent function, on the other hand, can be understood as the ratio of sine to cosine. The behavior and values of these functions repeat as you go around the unit circle, which leads to their periodic nature.

Tangent Limit

Understanding the limit of the tangent function, often written as \( \lim_{x \to a} \textrm{tan}\ x \), can pose some challenges, primarily because of the points where the function is undefined. A limit involving tangent is typically straightforward if it approaches a point where the function is continuous.

However, as the tangent function has vertical asymptotes — points where the function heads towards infinity — at \( \frac{\pi}{2} + k\pi \) for any integer k, it is essential to ensure the limit does not approach these values. The graph of \( \textrm{tan}\ x \) can help visualize this, showing a clear interruption at the asymptotes.

Unit Circle

The unit circle is a powerful tool in trigonometry and calculus for visualizing and understanding the behavior of trigonometric functions. It is a circle with a radius of one unit, centered at the origin of a coordinate plane.

The position of a point on the unit circle corresponds to the angle it makes with the positive x-axis, measured in radians. Each point on the unit circle can represent an angle, and the coordinates of that point are equal to the cosine and sine of that angle. In the context of limits, the unit circle allows us to visually determine the values of the sine and cosine functions, and by extension, the tangent function, for various angle approaches.