Skip to main content
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 2.6.10
Chapter 2, Problem 2.6.10

Evaluate f(3) if lim x→3^− f(x)=5,lim x→3^+ f(x)=6, and f is right-continuous at x=3.

Verified step by step guidance
1
Understand the concept of right-continuity: A function f is right-continuous at a point x=a if \( \lim_{x \to a^+} f(x) = f(a) \).
Identify the given information: \( \lim_{x \to 3^-} f(x) = 5 \), \( \lim_{x \to 3^+} f(x) = 6 \), and f is right-continuous at \( x=3 \).
Apply the definition of right-continuity at \( x=3 \): Since f is right-continuous at \( x=3 \), we have \( f(3) = \lim_{x \to 3^+} f(x) \).
Substitute the known limit from the right: \( f(3) = 6 \).
Conclude that the value of \( f(3) \) is determined by the right-hand limit, which is 6.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Limits

Limits describe the behavior of a function as the input approaches a certain value. In this case, we have left-hand and right-hand limits as x approaches 3, which are 5 and 6, respectively. Understanding limits is crucial for analyzing the continuity and behavior of functions at specific points.
Recommended video:
05:50
One-Sided Limits

Right-Continuity

A function is right-continuous at a point if the limit of the function as it approaches that point from the right equals the function's value at that point. In this scenario, since f is right-continuous at x=3, it implies that f(3) must equal the right-hand limit, which is 6.
Recommended video:
05:34
Intro to Continuity

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. In this question, the function f has different behaviors approaching from the left and right of x=3, which is a common scenario in piecewise definitions. Understanding how to evaluate such functions is essential for determining their values at specific points.
Recommended video:
05:36
Piecewise Functions
Related Practice
Textbook Question

Determine the following limits. 

lim θ→∞ cos θ / θ2

393
views
Textbook Question

Determine limx→∞f(x)\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\)) and limx→−∞f(x)\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\)) for the following functions. Then give the horizontal asymptotes of ff (if any).


f(x)=4x20x+1f\(\left\)(x\(\right\))=\(\frac{4x}{20x+1}\)

416
views
Textbook Question

Determine limx→∞f(x)\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\)) and limx→−∞f(x)\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\)) for the following functions. Then give the horizontal asymptotes of ff (if any).


f(x)=3x3−7x4+5x2f\(\left\)(x\(\right\))=\(\frac{3x^3-7}{x^4+5x^2}\)

338
views
Textbook Question

Find all vertical asymptotes x=ax=a of the following functions. For each value of aa, determine limx→a+f(x){\(\displaystyle\]\lim\)_{x\(\to\) a^{+}}}f\(\left\)(x\(\right\)), limx→a−f(x){\(\displaystyle\]\lim\)_{x\(\to\) a^{-}}}f\(\left\)(x\(\right\)), and limx→af(x){\(\displaystyle\]\lim\)_{x\(\to\) a}}f\(\left\)(x\(\right\)).

f(x)=cos(x)x2+2xf\(\left\)(x\(\right\))=\(\frac{\cos\left(x\right)}{x^2+2x}\)

348
views
Textbook Question

Determine the following limits.

lim x→3 1/ x − 3(1 /√x + 1 − 1/2)

398
views
Textbook Question

Determine the following limits. 


lim x→∞ x^4+7 / x^5+x^2−x

332
views