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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 9a
Chapter 2, Problem 9a

Let g(t)=t−9t−3g\(\left\)(t\(\right\))=\(\frac{t-9}{\sqrt{t}\)-3}.
Make two tables, one showing values of gg for t=8.9,8.99t=8.9,8.99, and 8.9998.999 and one showing values of gg for t=9.1,9.01t=9.1,9.01, and 9.0019.001.

Verified step by step guidance
1
Identify the function g(t) = \(\frac{t-9}{\sqrt{t}\)-3}. This function is undefined at t = 9 because both the numerator and the denominator become zero, leading to an indeterminate form.
To understand the behavior of g(t) as t approaches 9, we will evaluate the function for values of t slightly less than 9 and slightly greater than 9.
Create the first table for values of t less than 9: Calculate g(t) for t = 8.9, 8.99, and 8.999. For each value, substitute t into the function and simplify to find g(t).
Create the second table for values of t greater than 9: Calculate g(t) for t = 9.1, 9.01, and 9.001. Again, substitute each value of t into the function and simplify to find g(t).
Analyze the results from both tables to observe the trend of g(t) as t approaches 9 from both sides. This will help in understanding the limit of g(t) as t approaches 9.

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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 question, evaluating the function g(t) as t approaches 9 is crucial, as it may involve determining the limit to handle any indeterminate forms that arise.
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Rational Functions

A rational function is a ratio of two polynomials. The function g(t) = (t - 9) / (√t - 3) is a rational function, and understanding its behavior, especially around points where the denominator may approach zero, is essential for analyzing its values and limits.
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Table of Values

Creating a table of values involves calculating the output of a function for specific input values. In this question, constructing tables for g(t) at values close to 9 helps visualize the function's behavior and aids in understanding its limit and continuity around that point.
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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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