Math Problem Statement
Solution
Let's solve each problem from the image one by one.
1. Evaluate
This expression can be simplified using a trigonometric identity.
So, the expression becomes:
As approaches 0, . Therefore, the limit is:
2. Simplify the expression
This expression involves a limit that can be simplified by factoring the numerator.
Factor the numerator as:
Thus, the expression becomes:
Cancel out the common factor :
Now, substitute :
3. Evaluate the following limit
First, factor both the numerator and the denominator:
- Numerator:
- Denominator:
The expression becomes:
Cancel out the common factor :
Substitute :
4. Evaluate the limit \lim_{x \to 4} \frac{(x - 4)/(x^2 - x - 12)}
First, factor the quadratic expression in the denominator:
- Denominator:
The expression becomes:
Cancel out the common factor :
Substitute :
5. Evaluate the limit
For this limit, as , grows much slower than . Thus, the limit is:
Would you like to continue with the remaining problems or have any other questions?
Here are 5 related questions:
- How would the limit in problem 1 change if approaches ?
- What happens to the expression in problem 2 if we change the limit to ?
- How can L'Hôpital's rule be applied to problem 3?
- What does the expression in problem 4 evaluate to if the quadratic expression was not factorable?
- What would happen to the limit in problem 5 if we considered ?
Tip: Factoring and simplifying before taking limits often helps avoid indeterminate forms and simplifies the evaluation process.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Limits
Trigonometric identities
Factoring
L'Hôpital's rule
Formulas
-
Theorems
Limit properties
Suitable Grade Level
College level
Related Recommendation
Solving Various Limit Problems in Calculus
Step-by-Step Solutions for Trigonometric and Exponential Limit Problems
Step-by-Step Solutions for Exponential and Trigonometric Limits
Calculus Limit Problems Involving Exponential and Arctangent Functions
Solving Limit Problems with Indeterminate Forms and Trigonometric Functions