valuate this limit. 35x – 35 lim x-1 40x – 40 -
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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Question
![**Evaluating Limits in Calculus**
**5. Evaluate this limit:**
\[ \lim_{{x \to 1}} \frac{35x - 35}{40x - 40} \]
To solve this limit problem, we follow these steps:
1. **Simplify the expression:**
First, factor out the common terms in both the numerator and the denominator.
\[
\frac{35x - 35}{40x - 40} = \frac{35(x - 1)}{40(x - 1)}
\]
2. **Cancel common factors:**
Notice that \( (x - 1) \) is a common factor in both the numerator and the denominator, so we can cancel it out:
\[
\frac{35(x - 1)}{40(x - 1)} = \frac{35}{40}
\]
3. **Simplify the fraction:**
Simplify the fraction \(\frac{35}{40}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
\[
\frac{35}{40} = \frac{35 \div 5}{40 \div 5} = \frac{7}{8}
\]
4. **Conclusion:**
Therefore, the limit is:
\[
\lim_{{x \to 1}} \frac{35x - 35}{40x - 40} = \frac{7}{8}
\]
This simplified approach helps in understanding the process of evaluating limits by factoring and reducing fractions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F421794ac-cd89-4d48-bd79-93dc6e6c0071%2Fdb602fa3-fb2b-4803-8065-75efd7daa1d6%2Fa4nqifa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluating Limits in Calculus**
**5. Evaluate this limit:**
\[ \lim_{{x \to 1}} \frac{35x - 35}{40x - 40} \]
To solve this limit problem, we follow these steps:
1. **Simplify the expression:**
First, factor out the common terms in both the numerator and the denominator.
\[
\frac{35x - 35}{40x - 40} = \frac{35(x - 1)}{40(x - 1)}
\]
2. **Cancel common factors:**
Notice that \( (x - 1) \) is a common factor in both the numerator and the denominator, so we can cancel it out:
\[
\frac{35(x - 1)}{40(x - 1)} = \frac{35}{40}
\]
3. **Simplify the fraction:**
Simplify the fraction \(\frac{35}{40}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
\[
\frac{35}{40} = \frac{35 \div 5}{40 \div 5} = \frac{7}{8}
\]
4. **Conclusion:**
Therefore, the limit is:
\[
\lim_{{x \to 1}} \frac{35x - 35}{40x - 40} = \frac{7}{8}
\]
This simplified approach helps in understanding the process of evaluating limits by factoring and reducing fractions.
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