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 2.4.33
Chapter 2, Problem 2.4.33

Determine the following limits.


limx→0x3−5x2x2{\(\displaystyle\]\lim\)_{x\(\to\)0}}\(\frac{x^3-5x^2}{x^2}\)

Verified step by step guidance
1
Step 1: Identify the limit expression: \( \lim_{x \to 0} \frac{x^3 - 5x^2}{x^2} \).
Step 2: Simplify the expression by factoring out the common term in the numerator. The numerator \( x^3 - 5x^2 \) can be factored as \( x^2(x - 5) \).
Step 3: Cancel the common factor \( x^2 \) in the numerator and the denominator. This simplifies the expression to \( x - 5 \).
Step 4: Substitute \( x = 0 \) into the simplified expression \( x - 5 \) to evaluate the limit.
Step 5: Conclude the limit by substituting the value of \( x \) into the simplified expression.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
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 points of interest, including points where they may not be defined. For example, the limit of a function as x approaches 0 can reveal the function's value or behavior at that point, even if the function itself is not explicitly defined there.
Recommended video:
05:50
One-Sided Limits

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that can be multiplied together to obtain the original expression. In the context of limits, factoring can simplify complex rational expressions, making it easier to evaluate limits by canceling out common terms. For instance, in the limit problem given, factoring the numerator can help eliminate the indeterminate form that arises when substituting the limit directly.
Recommended video:
06:11
Limits of Rational Functions: Denominator = 0

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions that do not provide clear information about the limit's value, such as 0/0 or ∞/∞. These forms require further analysis, often through algebraic manipulation, L'Hôpital's Rule, or other techniques, to resolve. Recognizing an indeterminate form is crucial for applying the appropriate methods to find the actual limit.
Recommended video:
Guided course
3:56
Slope-Intercept Form
Related Practice
Textbook Question

Suppose f(x)→100 and g(x)→0, with g(x)<0 as x→2. Determine lim x→2 f(x) / g(x).

408
views
Textbook Question

Determine the following limits. 


lim x→∞ sin x / e^x

470
views
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.

323
views
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

365
views
Textbook Question

Determine the following limits.

lim x→∞ (x4 − 1) / (x5 + 2)

376
views
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

252
views