Problem 64 - Precalculus

Question

Given \(g(x)=\csc \left(x^{2}-17 x+52\right),\) find $$\lim _{x \rightarrow 13} g(x)$$

Step-by-Step Solution

Answer

The limit is infinite as the expression tends to approach zero at \( x = 13 \).

Step 1: Examine the function

The function given is \( g(x) = \csc(x^2 - 17x + 52) \). To analyze its behavior as \( x \to 13 \), we will first analyze the expression inside the cosecant function.

Step 2: Evaluate the quadratic inside the cosecant

Evaluate the expression inside the cosecant function at \( x = 13 \):\[ x^2 - 17x + 52 = 13^2 - 17 \times 13 + 52 = 169 - 221 + 52 \]Calculating further:\[ 169 - 221 + 52 = 0 \]

Step 3: Simplify the limit of the cosecant function

Since \( x^2 - 17x + 52 = 0 \) when \( x = 13 \), the expression becomes \( \csc(0) \), which is undefined, as cosecant is undefined at zero. Therefore, check how the function behaves near \( x = 13 \).

Step 4: Consider approaching from both sides

For \( x \to 13 \) from the left (\( x < 13 \)) and from the right (\( x > 13 \)), the quadratic expression \( x^2 - 17x + 52 \) approaches zero, leading \( \csc(x^2 - 17x + 52) \) to approach infinity. This indicates that the limit does not exist finitely.

Key Concepts

Trigonometric Limits

Trigonometric limits play a crucial role in various calculus problems, especially involving limits and continuity. These limits often involve trigonometric functions such as sine, cosine, tangent, and their reciprocals like cosecant, secant, and cotangent.

When dealing with trigonometric limits, it is essential to understand the behavior of trigonometric functions near specific points. In our problem, the focus is on the limit as \( x \to 13 \), where the expression inside the trigonometric function becomes zero.

The function often approaches an undefined value or exhibits a discontinuity at such points. Therefore, evaluating the behavior on either side of the point helps in understanding how the function behaves and whether a limit exists or not.

Cosecant Function

The cosecant function is the reciprocal of the sine function. It is defined as \( \csc(x) = \frac{1}{\sin(x)} \).

This means that wherever the sine function is zero, the cosecant is undefined. This happens at points like multiples of \( \pi \), such as \( 0, \pi, 2\pi, \ldots \).

In the given exercise, when evaluating \( g(x) = \csc(x^2 - 17x + 52) \) at \( x = 13 \), the expression inside the cosecant becomes zero, making the cosecant undefined. As a result, it is crucial to explore the function's behavior around \( x = 13 \). You check whether the function tends towards positive or negative infinity as it approaches this undefined point.

Undefined Limits

Undefined limits occur when a limit approaches an undefined value, such as infinity, or does not exist.

When dealing with an undefined limit, it often means the function values are becoming extremely large or small as the variable approaches a certain point.

In this problem, since the cosecant function approaches infinity when the quadratic expression tends to zero, \( \lim_{x \to 13} g(x) \) does not exist in a finite sense.

Often, examining both sides of the limit gives insight into the behavior of the function. This allows us to conclude that the limit tends toward infinity, thus remaining undefined at \( x = 13 \).