\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus - Limits Involving Inverse Trig Functions

Limits

Using Limits

Limits FAQs

Derivatives

Graphing

Related Rates

Optimization

Other Applications

Integrals

Improper Integrals

Trig Integrals

Length-Area-Volume

Applications - Tools

Infinite Series

Applications

Tools

Parametrics

Conics

Polar Coordinates

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Tools

Calculus Tools

Additional Tools

Articles

Limits

Using Limits

Limits FAQs

SV Calculus

MV Calculus

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Tools

Calculus Tools

Additional Tools

Articles

To find the limits of inverse trig functions, you need to be very familiar with how inverse trig functions work. Remember that \(\sin^{-1}(x) \neq 1/\sin(x)\) in this context. Go to the precalculus section on inverse trig functions if you need some review.

There are several ways to evaluate limits of inverse trig functions. Of course the most direct way is substitution, which you want to always try first. If that doesn't work, you can move the limit inside the inverse trig function like this.

\(\displaystyle{ \lim_{ x\to c }{ \arcsin(x) } = \arcsin \left[ \lim_{ x\to c }{ x } \right] }\)

This works with the other inverse functions as well since the inverse trig functions are continuous on their restricted domains. (Not sure what we mean? Review this page.)

Another idea is to convert to regular trig functions and try to see what will happen. So, for example, \(\theta = \arcsin(x) \to \sin(\theta) = x\). This will help since you are probably more familar with straight trig functions than you are with inverse trig functions. You may also be able to use trig identities to simplify the equation so that you can then evaluate the limit.

A third idea that will help you see what is going on is to graph the inverse trig function. You don't need to know the graphs off the top of your head since you probably already know the graphs of the trig functions. Just graph the trig functions and flip the graph about the line \(y=x\). In practice problem 2311, this is what he does. Although this technique is probably not enough work to get full credit for finding the limit, it can really help you visualize what is going on and may give you an idea on how to solve the problem.

Okay, try these practice problems.

Practice

Unless otherwise instructed, evaluate these limits. Give your answers in exact terms.

\(\displaystyle{ \lim_{x \to -1^+}{\sin^{-1}(x)} }\)

Problem Statement

Evaluate this limit. Give your answer in exact terms. \(\displaystyle{ \lim_{x \to -1^+}{\sin^{-1}(x)} }\)

Solution

2311 video

video by PatrickJMT

close solution

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{ \lim_{x \to \infty}{\arcsin\left[ \frac{1+x^2}{1+2x^2} \right] } }\)

Problem Statement

Evaluate this limit. Give your answer in exact terms. \(\displaystyle{ \lim_{x \to \infty}{\arcsin\left[ \frac{1+x^2}{1+2x^2} \right] } }\)

Solution

2312 video

video by PatrickJMT

close solution

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{ \lim_{x \to \infty}{\arctan(e^{3x})} }\)

Problem Statement

Evaluate this limit. Give your answer in exact terms. \(\displaystyle{ \lim_{x \to \infty}{\arctan(e^{3x})} }\)

Solution

2313 video

video by PatrickJMT

close solution

Log in to rate this practice problem and to see it's current rating.

\(\displaystyle{ \lim_{x \to 0^+}{ \arctan(\ln(x))} }\)

Problem Statement

Evaluate this limit. Give your answer in exact terms. \(\displaystyle{ \lim_{x \to 0^+}{ \arctan(\ln(x))} }\)

Solution

2314 video

video by PatrickJMT

close solution

Log in to rate this practice problem and to see it's current rating.

inverse trig limits 17calculus youtube playlist

Here is a playlist of the videos on this page.

You CAN Ace Calculus

Topics You Need To Understand For This Page

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

To bookmark this page and practice problems, log in to your account or set up a free account.

Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

math and science learning techniques

Get great tutoring at an affordable price with Chegg. Subscribe today and get your 1st 30 minutes Free!

The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

Practice

How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

Save Up To 50% Off SwissGear Backpacks Plus Free Shipping Over $49 at eBags.com!

Shop Amazon - Wearable Technology: Electronics

Page Sections

Save Up To 50% Off SwissGear Backpacks Plus Free Shipping Over $49 at eBags.com!

Prime Student 6-month Trial

Practice Instructions

Unless otherwise instructed, evaluate these limits. Give your answers in exact terms.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2020 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics
17Calculus
We use cookies to ensure that we give you the best experience on our website. By using this site, you agree to our Website Privacy Policy.