Question

Analyzing infinite limits graphically Graph the function \(y=\tan x\) with the window \([-\pi, \pi] \times[-10,10] .\) Use the graph to analyze the following limits. a. \(\lim _{x \rightarrow \pi / 2^{+}} \tan x\) b. \(\lim _{x \rightarrow \pi / 2^{-}} \tan x\) d. \(\quad \lim _{n}\) tan \(x\) c. \(\lim _{x \rightarrow-\pi / 2^{+}} \tan x\) d. \(\lim _{x \rightarrow-\pi / 2^{-}} \tan x\)

Short answer

a. \(\lim_{x \rightarrow \pi /2^+} \tan x\) b. \(\lim_{x \rightarrow \pi /2^-} \tan x\) c. \(\lim_{x \rightarrow -\pi /2^+} \tan x\) d. \(\lim_{x \rightarrow -\pi /2^-} \tan x\) Answer: a. \(\lim_{x \rightarrow \pi /2^+} \tan x = \infty.\) b. \(\lim_{x \rightarrow \pi /2^-} \tan x = -\infty.\) c. \(\lim_{x \rightarrow -\pi /2^+} \tan x = -\infty.\) d. \(\lim_{x \rightarrow -\pi /2^-} \tan x = \infty.\)

Step-by-step solution

Plot the function y=tan(x)

Graph the function \(y = \tan x\) using a graphing tool (e.g., graphing calculator, Desmos, etc.) with the window as \([-\pi, \pi] \times [-10, 10]\).

Analyze Limit A

We are asked to find \(\lim_{x \rightarrow \pi /2^+} \tan x\). Observe that as x approaches \(\pi /2\) from the right (or positive direction), the function's value shoots upward, approaching positive infinity. Therefore, the limit is: $$\lim_{x \rightarrow \pi /2^+} \tan x = \infty.$$

Analyze Limit B

We are asked to find \(\lim_{x \rightarrow \pi /2^-} \tan x\). Observe that as x approaches \(\pi /2\) from the left (or negative direction), the function's value shoots downward, approaching negative infinity. Therefore, the limit is: $$\lim_{x \rightarrow \pi /2^-} \tan x = -\infty.$$

Analyze Limit C

We are asked to find \(\lim_{x \rightarrow -\pi /2^+} \tan x\). Observe that as x approaches \(-\pi /2\) from the right (or positive direction), the function's value shoots downward, approaching negative infinity. Therefore, the limit is: $$\lim_{x \rightarrow -\pi /2^+} \tan x = -\infty.$$

Analyze Limit D

We are asked to find \(\lim_{x \rightarrow -\pi /2^-} \tan x\). Observe that as x approaches \(-\pi /2\) from the left (or negative direction), the function's value shoots upward, approaching positive infinity. Therefore, the limit is: $$\lim_{x \rightarrow -\pi /2^-} \tan x = \infty.$$ To summarize, after plotting the function \(y = \tan x\) in a graphing tool, we were able to analyze the limits graphically to obtain: a. \(\lim_{x \rightarrow \pi /2^+} \tan x = \infty.\) b. \(\lim_{x \rightarrow \pi /2^-} \tan x = -\infty.\) c. \(\lim_{x \rightarrow -\pi /2^+} \tan x = -\infty.\) d. \(\lim_{x \rightarrow -\pi /2^-} \tan x = \infty.\)

Key concepts

Graphical Analysis

Graphical analysis involves studying the visual representation of mathematical functions. When we plot a graph, it helps us understand how the function behaves for different values of \(x\).

In this exercise, we use a specific window, \([-\pi, \pi] \times [-10, 10]\), to plot \(y = \tan x\). This graph shows the repeating pattern or periodicity of the tangent function.

• Key features like peaks, valleys, and asymptotes become visible.
• Observing these features helps us deduce the infinite nature of some limits.

Graphical analysis makes it easier to interpret infinite limits since numerical approaches might struggle with very large or very small numbers.

Trigonometric Functions

Trigonometric functions like \(\tan x\) are essential in both geometry and calculus. The tangent function specifically represents the ratio of the sine and cosine functions.

• The function \(y = \tan x\) is defined as \(\tan x = \frac{\sin x}{\cos x}\).
• It has a period of \(\pi\), meaning it repeats every \(\pi\) units.

At certain points, such as \(x = \frac{\pi}{2}\) and \(x = -\frac{\pi}{2}\), the tangent function becomes undefined. This happens because \(\cos x\) becomes zero, leading to division by zero situations. Trigonometric functions thus have unique properties that affect how they behave around certain points.

Limits

Understanding limits is a core concept in calculus, describing the behavior of functions as they approach specific points. Limits help determine how a function behaves near points where it might be undefined.

For the tangent function, infinite limits occur as \(x\) approaches points where \(\tan x\) is undefined, like \(\frac{\pi}{2}\) and \(-\frac{\pi}{2}\).

• As \(x \rightarrow \frac{\pi}{2}\) from the right, \(\tan x\) approaches \(\infty\).
• As \(x \rightarrow \frac{\pi}{2}\) from the left, \(\tan x\) heads towards \(-\infty\).

These infinite limits demonstrate how close examination of behavior near points of interest can reveal more about the function's overall pattern.

Vertical Asymptotes

Vertical asymptotes represent lines that a function approaches but never quite reaches or crosses. They occur in functions that become unbounded.

For \(y = \tan x\), vertical asymptotes occur at points where the cosine function equals zero, causing the tangent function to tend towards infinity or negative infinity.

• In our exercise, these asymptotes happen at \(x = \frac{\pi}{2}\) and \(x = -\frac{\pi}{2}\).
• The graphs of these functions will show lines that direct towards those unbounded values.

Understanding vertical asymptotes is crucial for graph interpretation and recognizing trends in trigonometric functions.