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 2.29
Chapter 2, Problem 2.29

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim x→3 −5x / √4x − 3

Verified step by step guidance
1
Identify the limit expression: \( \lim_{{x \to 3}} \frac{{-5x}}{{\sqrt{{4x}} - 3}} \).
Substitute \( x = 3 \) into the expression to check if it results in an indeterminate form.
Calculate \( \sqrt{{4 \times 3}} - 3 \) to see if the denominator becomes zero.
If the expression is indeterminate, consider rationalizing the denominator or using L'Hôpital's Rule if applicable.
Evaluate the limit using the chosen method to simplify the expression.

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

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for evaluating continuity and differentiability. In this case, we are interested in the limit of the function as x approaches 3.
Recommended video:
05:50
One-Sided Limits

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit different behaviors depending on the values of x, particularly at points where the denominator is zero. Understanding how to simplify and analyze these functions is essential for finding limits, especially when approaching points that may lead to indeterminate forms.
Recommended video:
6:04
Intro to Rational Functions

Indeterminate Forms

Indeterminate forms occur when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is crucial, as they often require additional techniques, such as L'Hôpital's Rule or algebraic manipulation, to resolve and find the actual limit.
Recommended video:
Guided course
3:56
Slope-Intercept Form
Related Practice
Textbook Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

f(x)=sinxf\(\left\)(x\(\right\))=\(\sin\) x

Textbook Question

Estimate the following limits using graphs or tables.


lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4

Textbook Question

Use the definitions given in Exercise 57 to prove the following infinite limits.


lim x→1^+ 1 /1 − x=−∞

Textbook Question

Let g(x)={x2+xif x<1aif x=13x+5if x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)x^2+x & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ 3x+5 & \(\text{if }\)x>1\(\end{cases}\)

a. Determine the value of a for which gg is continuous from the left at 11.

Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→7 f(x)=9, where f(x)={3x−12 if x≤7

x+2 if x>7

Textbook Question

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 


f(x)=(2x−3)^2/3