Question

Investigate the following limits. $$\lim _{\theta \rightarrow \pi / 2^{+}} \frac{1}{3} \tan \theta$$

Short answer

Answer: The limit of the given function as θ approaches π/2 from the positive side is infinity.

Step-by-step solution

Observe the function at the limit point

Notice that at θ = π/2, the tangent function, tan(θ), is undefined because the denominator (cos(θ)) becomes zero. Therefore, the limit involves the tangent function approaching infinity.

Behavior of the tangent function near π/2

As θ approaches π/2 from the positive side, the tangent function increases without bound, approaching infinity. Therefore, we can describe the behavior of the tangent function near π/2 as: $$\lim _{\theta \rightarrow \pi / 2^{+}} \tan \theta = \infty$$

Multiply by the constant

Now that we know the behavior of the tangent function near π/2, we can multiply it by the constant (1/3). It is known that multiplying a constant by infinity still results in infinity. So, the limit of the given function is: $$\lim _{\theta \rightarrow \pi / 2^{+}} \frac{1}{3} \tan \theta = \frac{1}{3} \infty = \infty$$ So the limit of the given function as θ approaches π/2 from the positive side is infinity.

Key concepts

Tangent Function Behavior

The tangent function, represented as \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), displays unique behavior that is crucial to understanding limits involving this function. As \(\theta\) approaches specific values, particularly at odd multiples of \(\frac{\pi}{2}\) like \(\frac{\pi}{2}\), \(\tan(\theta)\) experiences undefined points due to the cosine in the denominator approaching zero. This causes the tangent function to increase or decrease without bound, illustrating a vertical asymptote on the graph of the function. The notion that \(\tan(\theta)\) approaches \(\infty\) or \(-\infty\) near these points is why it is often used in understanding behavior around critical values in calculus. By visualizing or graphing the function, you can see how the tangent line behaves just before reaching the vertical asymptotic boundary, increasing indefinitely.

Undefined Limits

An undefined limit occurs when a function does not approach a finite value as the input nears a specific point. This often happens with functions like the tangent function at its vertical asymptotes. As \(\theta\) moves toward \(\frac{\pi}{2}\) from the positive side, \(\tan(\theta)\) doesn't settle at any numerical value, but instead races toward infinity. This creates what we call an undefined limit.

• These limits do not give a fixed number.
• They illustrate scenarios where function values can become infinitely large or small.

Knowing when limits are undefined is key in calculus as it highlights points of discontinuity or behavioral changes in a function's graph. Rather than approaching any specific value, the function diverges, which is an essential idea in many advanced calculus problems.

Asymptotic Behavior

Understanding asymptotic behavior is vital in analyzing how functions behave as the input approaches certain critical points. Asymptotes are lines that a graph approaches but never actually touches or crosses. In the case of the tangent function, we observe vertical asymptotes at odd multiples of \(\frac{\pi}{2}\). This means:

• As \(\theta\) gets closer to \(\frac{\pi}{2}\) from the right (\(\theta \rightarrow \frac{\pi}{2}^{+}\)), \(\tan(\theta)\) climbs to infinity, never leveling off or touching the line \(\theta = \frac{\pi}{2}\).
• The graph exhibits asymptotic behavior by appearing to shoot off into the sky, representing values stretching indefinitely without ever crossing the vertical boundaries.

Asymptotic analysis helps us predict how functions behave at infinity or bound towards infinite limits, offering insight into function growth and behavior in applied contexts or geometric interpretations.