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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 11
Chapter 2, Problem 11

Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.


lim x→1 f(x) / g(x)−h(x)

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Identify the given limits: \( \lim_{{x \to 1}} f(x) = 8 \), \( \lim_{{x \to 1}} g(x) = 3 \), and \( \lim_{{x \to 1}} h(x) = 2 \).
Recognize that the problem requires finding \( \lim_{{x \to 1}} \frac{f(x)}{g(x) - h(x)} \).
Apply the limit law for quotients: \( \lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{\lim_{{x \to a}} f(x)}{\lim_{{x \to a}} g(x)} \) provided \( \lim_{{x \to a}} g(x) \neq 0 \).
Calculate the limit of the denominator: \( \lim_{{x \to 1}} (g(x) - h(x)) = \lim_{{x \to 1}} g(x) - \lim_{{x \to 1}} h(x) = 3 - 2 = 1 \).
Use the quotient limit law: \( \lim_{{x \to 1}} \frac{f(x)}{g(x) - h(x)} = \frac{\lim_{{x \to 1}} f(x)}{\lim_{{x \to 1}} (g(x) - h(x))} = \frac{8}{1} \).

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Key Concepts

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

Limit of a Function

The limit of a function describes the value that a function approaches as the input approaches a certain point. In this case, we have limits for functions f(x), g(x), and h(x) as x approaches 1. Understanding limits is crucial for evaluating expressions involving functions at specific points.
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Limits of Rational Functions: Denominator = 0

Limit Laws

Limit laws are rules that allow us to compute the limits of functions based on the limits of their components. For example, the quotient law states that if the limits of f(x) and g(x) exist and g(x) is not zero at the limit point, then lim x→c (f(x)/g(x)) = lim x→c f(x) / lim x→c g(x). These laws are essential for simplifying and calculating complex limits.
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One-Sided Limits

Algebraic Manipulation of Limits

Algebraic manipulation involves rearranging and simplifying expressions to facilitate limit evaluation. In the given problem, we need to compute the limit of the expression f(x)/g(x) - h(x). This requires applying limit laws and performing algebraic operations to find the limit of the entire expression as x approaches 1.
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Finding Limits by Direct Substitution
Related Practice
Textbook Question

Let g(t)=t−9t−3g\(\left\)(t\(\right\))=\(\frac{t-9}{\sqrt{t}\)-3}.

Make a conjecture about the value of limt→9t−9t−3{\(\displaystyle\]\lim\)_{t\(\to\)9}\(\frac{t-9}{\sqrt{t}\)-3}}.

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Textbook Question

Determine the following limits.

lim x→1 √5x+6

Textbook Question

Determine the following limits.

lim x→1000 18π^2

Textbook Question

Given the function f(x)=−16x2+64xf\(\left\)(x\(\right\))=-16x^2+64x, complete the following. <IMAGE>

Find the slopes of the secant lines that pass though the points (x,f(x))\(\left\)(x,f\(\left\)(x\(\right\))\(\right\)) and (2,f(2))\(\left\)(2,f\(\left\)(2\(\right\))\(\right\)), for x=1.5,1.9,1.99,1.999,x=1.5,1.9,1.99,1.999, and 1.99991.9999 (see figure).

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Textbook Question

Determine the following limits.

lim h→0 √5x + 5h − √5x / h, where x>0 Is constant

Textbook Question

Given the function f(x)=−16x2+64xf\(\left\)(x\(\right\))=-16x^2+64x, complete the following. <IMAGE>

Make a conjecture about the value of the limit of the slopes of the secant lines that pass through (x,f(x))\(\left\)(x,f\(\left\)(x\(\right\))\(\right\)) and (2,f(2))\(\left\)(2,f\(\left\)(2\(\right\))\(\right\)) as xx approaches 22.