/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Find the limits. \(f(x)=4-x^{2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 19

Find the limits. \(f(x)=4-x^{2}, g(x)=\sqrt{x+1}\) (a) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 3} g(x)\) (c) \(\lim _{x \rightarrow 1} g(f(x))\)

Short Answer

Expert verified
\(\lim _{x \rightarrow 1} f(x) = 3, \lim _{x \rightarrow 3} g(x) = 2, \lim _{x \rightarrow 1} g(f(x)) = 2\)

Step by step solution

01

Evaluate \(\lim _{x \rightarrow 1} f(x)\)

Substitute \(x = 1\) into \(f(x)\) to find \(\lim _{x \rightarrow 1} f(x)\). As a result, \(f(1) = 4 - 1^{2} = 3\). Therefore, \(\lim _{x \rightarrow 1} f(x) = 3\)
02

Evaluate \(\lim _{x \rightarrow 3} g(x)\)

Substitute \(x = 3\) into \(g(x)\) to find \(\lim _{x \rightarrow 3} g(x)\). Accordingly, \(g(3) = \sqrt{3 + 1} = \sqrt{4} = 2\). Thus, \(\lim _{x \rightarrow 3} g(x) = 2\)
03

Evaluate \(\lim _{x \rightarrow 1} g(f(x))\)

Substitute \(x = 1\) into \(f(x)\) first, then substitute this output into \(g(x)\) to find \(\lim _{x \rightarrow 1} g(f(x))\). Specifically, \(f(1) = 3\) (from step 1) and substituting this into \(g(x)\), \(g(3) = 2\) (from step 2). Therefore, \(\lim _{x \rightarrow 1} g(f(x)) = 2\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

$$ \begin{aligned} &\text { Prove that if } f \text { and } g \text { are one-to-one functions, then }\\\ &(f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x). \end{aligned} $$

Rate of Change A 25 -foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate \(r\) of \(r=\frac{2 x}{\sqrt{625-x^{2}}} \mathrm{ft} / \mathrm{sec}\) where \(x\) is the distance between the ladder base and the house. (a) Find \(r\) when \(x\) is 7 feet. (b) Find \(r\) when \(x\) is 15 feet. (c) Find the limit of \(r\) as \(x \rightarrow 25^{-}\).

Write the expression in algebraic form. \(\sec [\arcsin (x-1)]\)

Write the expression in algebraic form. \(\sin (\arccos x)\)

Explain why the function has a zero in the given interval. $$ \begin{array}{lll} \text { Function } & \text { Interval } \\ \hline f(x)=x^{2}-4 x+3 & {[2,4]} \\ \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.