lim(-x² + x-2) x-2 x+5x+4 lim x+2 2 +7x +6 lim I+3
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![Below are four limit problems that illustrate different types of indeterminate forms and techniques used to solve limits. These are commonly encountered in a calculus course and are essential for understanding continuity and the behavior of functions as they approach specific points.
1. **Problem 1:**
\[
\lim_{{x \to 2}} (-x^2 + x - 2)
\]
- This limit involves a polynomial function. To find the limit as \(x\) approaches 2, evaluate the function at \(x = 2\).
2. **Problem 2:**
\[
\lim_{{x \to 4}} \frac{{x^2 + 5x + 4}}{{x + 2}}
\]
- This problem involves a rational function. The limit can be simplified by factoring the numerator and checking for any possible simplifications before directly substituting \(x = 4\).
3. **Problem 3:**
\[
\lim_{{x \to 0}} \frac{{x^2 + 7x + 6}}{{x + 3}} =
\]
- Another rational function, but here you need to factor the numerator and see if it simplifies with the denominator to find the limit as \(x\) approaches 0.
4. **Problem 4:**
\[
\lim_{{x \to -3}} \frac{{x^2 + x - 6}}{{x^2 + 8x + 15}}
\]
- This involves both the numerator and denominator being quadratic polynomials. Factoring both can help in simplifying the expression to evaluate the limit as \(x\) approaches -3.
In summary, these problems require knowledge of factoring polynomials, simplifying rational expressions, and direct substitution. In cases where direct substitution results in indeterminate forms such as \(\frac{0}{0}\), further algebraic manipulation or applying L'Hôpital's rule may be necessary.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3263f26b-dafd-4af0-9fa9-4c675e54fa32%2Ffa1d1217-4ed4-428f-b961-da3124804991%2F26wy7k6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Below are four limit problems that illustrate different types of indeterminate forms and techniques used to solve limits. These are commonly encountered in a calculus course and are essential for understanding continuity and the behavior of functions as they approach specific points.
1. **Problem 1:**
\[
\lim_{{x \to 2}} (-x^2 + x - 2)
\]
- This limit involves a polynomial function. To find the limit as \(x\) approaches 2, evaluate the function at \(x = 2\).
2. **Problem 2:**
\[
\lim_{{x \to 4}} \frac{{x^2 + 5x + 4}}{{x + 2}}
\]
- This problem involves a rational function. The limit can be simplified by factoring the numerator and checking for any possible simplifications before directly substituting \(x = 4\).
3. **Problem 3:**
\[
\lim_{{x \to 0}} \frac{{x^2 + 7x + 6}}{{x + 3}} =
\]
- Another rational function, but here you need to factor the numerator and see if it simplifies with the denominator to find the limit as \(x\) approaches 0.
4. **Problem 4:**
\[
\lim_{{x \to -3}} \frac{{x^2 + x - 6}}{{x^2 + 8x + 15}}
\]
- This involves both the numerator and denominator being quadratic polynomials. Factoring both can help in simplifying the expression to evaluate the limit as \(x\) approaches -3.
In summary, these problems require knowledge of factoring polynomials, simplifying rational expressions, and direct substitution. In cases where direct substitution results in indeterminate forms such as \(\frac{0}{0}\), further algebraic manipulation or applying L'Hôpital's rule may be necessary.
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