Textbook Question
Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.
lim x →π cos² (x― tan x)
183
views
1. Limits and Continuity
Finding Limits Algebraically
2:35 minutesProblem 2.2.25
Textbook Question
Limits of quotients
Find the limits in Exercises 23–42.
limx→−5 (x² + 3x − 10) / x + 5
Verified step by step guidance1
First, identify the type of limit problem. This is a limit of a quotient as x approaches -5.
Check if direct substitution of x = -5 into the expression (x² + 3x − 10) / (x + 5) results in an indeterminate form like 0/0.
Substitute x = -5 into the numerator: x² + 3x − 10 becomes (-5)² + 3(-5) − 10.
Substitute x = -5 into the denominator: x + 5 becomes -5 + 5.
If the substitution results in 0/0, factor the numerator x² + 3x − 10 to see if it can be simplified with the denominator x + 5. Look for common factors to cancel out and resolve the indeterminate form.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey 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. They help in understanding the behavior of functions near specific points, especially when direct substitution may lead to indeterminate forms. In this case, evaluating the limit as x approaches -5 requires analyzing the function's behavior around that point.
Recommended video:
05:50
One-Sided Limits
Quotient of Functions
The quotient of functions involves dividing one function by another. When finding limits of quotients, it is essential to consider the behavior of both the numerator and denominator as the variable approaches a specific value. If the denominator approaches zero while the numerator does not, the limit may be undefined or infinite, necessitating further analysis or simplification.
Recommended video:
06:43The Quotient Rule
Factoring and Simplifying
Factoring and simplifying expressions is a crucial technique in calculus, especially when dealing with limits. By factoring the numerator and denominator, one can often cancel common terms, which can resolve indeterminate forms like 0/0. In this problem, factoring the numerator will help in simplifying the expression before evaluating the limit as x approaches -5.
Recommended video:
6:36Simplifying Trig Expressions
Related Videos
Related Practice