Problem 15 - Calculus with Concepts in Calculus

Question

Find the given limit. $$ \lim _{x \rightarrow \infty} \tan \frac{1}{x} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Expression Inside the Limit

We need to find \( \lim_{x \rightarrow \infty} \tan \frac{1}{x} \). First, consider the expression inside the tangent function: \( \frac{1}{x} \). As \( x \) approaches infinity, \( \frac{1}{x} \) approaches 0 because dividing 1 by an increasingly larger number results in a value close to 0.

Step 2: Consider the Limit of the Inner Function

Since \( \frac{1}{x} \rightarrow 0 \) as \( x \rightarrow \infty \), we need to evaluate \( \tan(\text{expression}) \) as the expression approaches 0. Since the tangent function is continuous around 0, computing the limit of \( \tan \frac{1}{x} \) simplifies to finding the tangent of the limit we found.

Step 3: Apply Continuity of the Tangent Function

The tangent function is continuous at 0, meaning \( \tan \) of a value approaching 0 is \( \tan(0) \). Thus, \( \lim_{x \rightarrow \, \infty} \tan \frac{1}{x} = \tan(0) \).

Step 4: Evaluate \( \tan(0) \)

The value \( \tan(0) \) is 0 because the tangent of 0 degrees (or 0 radians) is 0. Hence, \( \lim_{x \rightarrow \, \infty} \tan \frac{1}{x} = 0 \).

Key Concepts

Continuity in Calculus

Continuity is a vital concept in calculus, especially when determining limits. For a function to be continuous at a point, the limit as the variable approaches that point must exist, and it must equal the function's value at that point. This means there are no breaks, jumps, or holes in the function around that point.
When evaluating limits, such as the one given in the exercise, continuity can simplify the process significantly. If the function in question is continuous at a point where the input approaches, then the limit of the function as it approaches that point can be directly calculated by substituting the approaching value into the function.
In our original exercise, we deal with the tangent function, which is continuous around 0. This property allowed us to calculate the limit directly by substituting the approaching value since there are no abrupt changes in the tangent function's value near 0.

Understanding Infinity in Limits

Infinity in calculus often describes behaviors as values grow indefinitely large. It’s not a number but a concept depicting unbounded growth. When variables approach infinity, they tend to give insight into the end behavior of functions.
In the given problem, as the variable\[ x \rightarrow \infty \]it means we’re observing the behavior of the function as numbers get larger without bounds. This informs us that\[ \frac{1}{x} \rightarrow 0 \]as x becomes perpetually large because the quotient of a consistent number and a growing number dwindles towards zero.
Understanding this movement helps simplify finding the limit, because infinity-related questions often set the stage for substitution in continuous functions like the tangent function.

Tangent Function and Its Properties

The tangent function, represented by \( \tan(x) \), is a trigonometric function showing the ratio of the sine to cosine function. Unlike other trigonometric functions like sine and cosine that oscillate between -1 and 1, tangent can output any real number, making it unique.
The tangent function is periodic with a period of \( \pi \), and it’s known for its asymptotes wherever \( x = \frac{(2n+1)\pi}{2} \), meaning it’s undefined at these points.
At values near these asymptotes, the function spikes up or down to infinity, but around 0, it behaves nicely and stays continuous. This property aligns perfectly with our given problem since it’s crucial to note that the tangent function is specifically continuous at 0.

• At 0, \( \tan(0) = 0 \), rendering straightforward evaluations for limits approaching zero.
• In practical applications, knowing the continuity and behavior around important points like 0 can simplify finding limits like the one in our exercise.