Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?
lim x → π/6 √(csc² x + 5√3 tan x)

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Identify the function for which you need to find the limit: \( \sqrt{\csc^2 x + 5\sqrt{3} \tan x} \). You are approaching the point \( x = \frac{\pi}{6} \).

Recall the trigonometric identities: \( \csc x = \frac{1}{\sin x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Use these identities to express \( \csc^2 x \) and \( \tan x \) in terms of \( \sin x \) and \( \cos x \).

Substitute \( x = \frac{\pi}{6} \) into the trigonometric functions: \( \sin \frac{\pi}{6} = \frac{1}{2} \) and \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \). Calculate \( \csc^2 \frac{\pi}{6} = \left(\frac{1}{\sin \frac{\pi}{6}}\right)^2 \) and \( \tan \frac{\pi}{6} = \frac{\sin \frac{\pi}{6}}{\cos \frac{\pi}{6}} \).

Substitute these values into the expression \( \sqrt{\csc^2 x + 5\sqrt{3} \tan x} \) to evaluate the limit as \( x \to \frac{\pi}{6} \).

Determine if the function is continuous at \( x = \frac{\pi}{6} \) by checking if the limit equals the function value at that point. If \( \lim_{x \to \frac{\pi}{6}} f(x) = f\left(\frac{\pi}{6}\right) \), then the function is continuous at \( x = \frac{\pi}{6} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this case, we are interested in the limit of the function as x approaches π/6. Understanding limits helps in analyzing the behavior of functions near specific points, which is crucial for determining continuity and differentiability.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For the given limit, we need to check if the function is defined at x = π/6 and if the limit equals the function's value there. Continuity ensures that there are no breaks, jumps, or holes in the graph of the function at that point.

Trigonometric Functions

Trigonometric functions, such as csc (cosecant) and tan (tangent), are essential in calculus for analyzing periodic behavior and relationships in triangles. In this limit, we need to evaluate csc² x and tan x at x = π/6. Understanding their values and properties is crucial for simplifying the expression and accurately finding the limit.

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