Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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1:35 minutes

Problem 51d

Textbook Question

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.
lim x→π/2^− tan x

Verified step by step guidance

Step 1: Understand the behavior of the tangent function. The function $ y = \tan x $ has vertical asymptotes where the cosine of $ x $ is zero, which occurs at $ x = \frac{\pi}{2} + k\pi $ for any integer $ k $.

Step 2: Identify the limit direction. The limit $ \lim_{x \to \frac{\pi}{2}^-} \tan x $ means we are approaching $ \frac{\pi}{2} $ from the left side, or from values less than $ \frac{\pi}{2} $.

Step 3: Analyze the behavior of $ \tan x $ as $ x $ approaches $ \frac{\pi}{2} $ from the left. As $ x $ approaches $ \frac{\pi}{2}^- $, $ \tan x $ increases without bound because the cosine of $ x $ approaches zero from the positive side, making $ \tan x = \frac{\sin x}{\cos x} $ approach positive infinity.

Step 4: Sketch the graph of $ y = \tan x $ over the interval $[-\pi, \pi]$. Note the vertical asymptotes at $ x = -\frac{\pi}{2} $ and $ x = \frac{\pi}{2} $, and the periodic nature of the tangent function with period $ \pi $.

Step 5: Use the graph to verify the limit. On the graph, observe that as $ x $ approaches $ \frac{\pi}{2} $ from the left, the value of $ \tan x $ increases towards positive infinity, confirming the limit analysis.

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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 tangent function as x approaches π/2 from the left. Understanding limits helps in analyzing the behavior of functions near points of discontinuity or asymptotes.

Tangent Function

The tangent function, defined as tan(x) = sin(x)/cos(x), is periodic and has vertical asymptotes where the cosine function equals zero, such as at x = π/2. This characteristic leads to the function approaching infinity as x approaches these points from the left or right, which is crucial for evaluating the limit in the question.

Graphing Functions

Graphing functions provides a visual representation of their behavior, including limits and asymptotes. By sketching the graph of y = tan(x) over the specified window, one can observe the function's approach to infinity as x nears π/2, confirming the analytical limit found. This visual check enhances understanding of the function's properties.

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Related Practice

Textbook Question

Postage rates Assume postage for sending a first-class letter in the United States is $0.47 for the first ounce (up to and including 1 oz) plus $0.21 for each additional ounce (up to and including each additional ounce).

a. Graph the function p=f(w) that gives the postage p for sending a letter that weighs w ounces, for 0<w≤3.5.

Textbook Question

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^+ tan x

Textbook Question

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^− tan x

Textbook Question

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^+ tan x

Textbook Question

Find polynomials p and q such that f=p/q is undefined at 1 and 2, but f has a vertical asymptote only at 2. Sketch a graph of your function.

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

The line x=−1 is a vertical asymptote of the function f(x) =x^2 − 7x + 6 / x^2 − 1.

Textbook Question

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

f(x)=x^2−3x+2 / x^10−x^9

Textbook Question

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

h(x)=e^x(x+1)^3

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Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.lim x→π/2^− tan x

Key Concepts

Limits

Tangent Function

Graphing Functions

Watch next

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.
lim x→π/2^− tan x