Skip to main content
Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.5.36
Chapter 2, Problem 2.5.36

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)

Verified step by step guidance
1
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 \( x = \frac{\pi}{6} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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.
Recommended video:
05:50
One-Sided Limits

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.
Recommended video:
05:34
Intro to Continuity

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.
Recommended video:
6:04
Introduction to Trigonometric Functions
Related Practice
Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limt→0 2t / tan t

Textbook Question

Limits and Continuity

Repeat the instructions of Exercise 1 for


1 , x ≤ ―1

1/x , 0 < |x| < 1

ƒ(x) = { 0, x = 1 ,

1 , x > 1 .

Textbook Question

Centering Intervals About a Point


In Exercises 1–6, sketch the interval (a,b), on the x-axis with the point c inside. Then find a value of δ>0 such that a < x < b whenever 0 < |x−c| < δ.


a=4/9, b=4/7, c=1/2

Textbook Question

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx→3 (3x − 7) = 2

Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limx→0 (1 − cos 3x) / 2x

Textbook Question

Finding Limits of Differences When x → ±∞


Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)


lim x → ∞ (√(9x² − x) − 3x)