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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 2.6.30
Chapter 2, Problem 2.6.30

Determine the interval(s) on which the following functions are continuous. 
f(t)=t+2 / t^2−4

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The function \( f(t) = \frac{t+2}{t^2-4} \) is a rational function, which is continuous everywhere in its domain.
A rational function is undefined where its denominator is zero. Set the denominator equal to zero: \( t^2 - 4 = 0 \).
Factor the equation: \( (t-2)(t+2) = 0 \). This gives the solutions \( t = 2 \) and \( t = -2 \).
The function is undefined at \( t = 2 \) and \( t = -2 \). Therefore, these points are not in the domain of the function.
The function is continuous on the intervals \((-\infty, -2)\), \((-2, 2)\), and \((2, \infty)\).

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Key Concepts

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

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. This means there are no breaks, jumps, or asymptotes in the function's graph.
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Intro to Continuity

Identifying Discontinuities

Discontinuities in a function can occur due to points where the function is undefined, such as division by zero. For the function f(t) = (t + 2) / (t^2 - 4), we need to find values of t that make the denominator zero, as these will indicate points of discontinuity. Factoring the denominator helps identify these critical points.
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Intro to Continuity Example 1

Intervals of Continuity

Once discontinuities are identified, the next step is to determine the intervals of continuity. This involves analyzing the real number line and excluding the points of discontinuity to find continuous segments. The function is continuous on intervals that do not include these points, which can be expressed in interval notation.
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Intro to Continuity Example 1
Related Practice
Textbook Question

Consider the position function s(t) =−16t^2+100t representing the position of an object moving vertically along a line. Sketch a graph of s with the secant line passing through (0.5, s(0.5)) and (2, s(2)). Determine the slope of the secant line and explain its relationship to the moving object.

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Textbook Question

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

f(x)=1/ √x sec x

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Textbook Question

Evaluate each limit. 


limx→π2sin(x)−1sin(x)−1{\(\displaystyle\[\lim\)_{x\(\to\]\frac{\pi}{2}\)}{\(\frac{\sin\left(x\right)-1}{\sqrt{\sin\left(x\right)}\)-1}}}

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Textbook Question

Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all the following conditions.


g(2) =1,g(5) =−1,lim x→4 g(x) =−∞,lim x→7^− g(x) =∞,lim x→7^+ g(x) =−∞

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Textbook Question

A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,…} is specified by the function f(n) = 2n, where n=1,2,3,….The limit of such a sequence is lim n→∞ f(n), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. 


{0,1/2,2/3,3/4,…}, which is defined by f(n) = (n−1) / n, for n=1,2,3,…

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Textbook Question

Use an appropriate limit definition to prove the following limits.


lim x→ 5x^2 − 25 / x − 5=10

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