Evaluate the limit: limit, start subscript, x, right arrow, 0, end subscript, start fraction, 5, x, minus, 5, x, cosine, left parenthesis, x, right parenthesis, divided by, 7, x, sine, left parenthesis, x, right parenthesis, end fraction 
x→0
lim
​
  
7xsin(x)
5x−5xcos(x)
​
 


To evaluate the limit 

\[
\lim_{x \to 0} \frac{5x - 5x \cos(x)}{7x \sin(x)},
\]

let’s simplify the expression step-by-step.

### Step 1: Factor out \( x \) in the numerator and denominator
Rewrite the expression by factoring \( x \) from both the numerator and the denominator:

\[
\frac{5x - 5x \cos(x)}{7x \sin(x)} = \frac{5x(1 - \cos(x))}{7x \sin(x)}.
\]

Now, cancel \( x \) from the numerator and the denominator:

\[
= \frac{5(1 - \cos(x))}{7 \sin(x)}.
\]

### Step 2: Apply limit properties and trigonometric identities
Now we can directly substitute the limit \( x \to 0 \):

1. **For \( 1 - \cos(x) \):** We know that \( 1 - \cos(x) \approx \frac{x^2}{2} \) as \( x \to 0 \).
2. **For \( \sin(x) \):** We also know that \( \sin(x) \approx x \) as \( x \to 0 \).

Using these approximations, substitute into the fraction:

\[
\frac{5 \cdot \frac{x^2}{2}}{7 \cdot x} = \frac{\frac{5x^2}{2}}{7x}.
\]

### Step 3: Simplify the fraction
Now, divide both the numerator and denominator by \( x \):

\[
= \frac{\frac{5x}{2}}{7} = \frac{5x}{14}.
\]

### Step 4: Take the limit as \( x \to 0 \)
As \( x \to 0 \), the term \( \frac{5x}{14} \to 0 \).

### Conclusion
Thus,

\[
\lim_{x \to 0} \frac{5x - 5x \cos(x)}{7x \sin(x)} = 0.
\]

Would you like more details on each step or have any questions?

---

### Related Questions
1. How do trigonometric approximations like \( \sin(x) \approx x \) help in limit problems?
2. What happens if we use L'Hôpital's Rule to solve this limit?
3. Can we use series expansions for more complex trigonometric limits?
4. Why does \( 1 - \cos(x) \approx \frac{x^2}{2} \) as \( x \to 0 \)?
5. How can this problem be adapted to use polar coordinates?

### Tip
In trigonometric limits, approximations are often essential. Familiarize yourself with common ones like \( \sin(x) \approx x \) and \( 1 - \cos(x) \approx \frac{x^2}{2} \) for faster problem-solving.

Evaluate the limit: lim_{x→0} (5x - 5x cos(x)) / (7x sin(x))

Explore the step-by-step solution for evaluating the limit of (5x - 5x cos(x)) / (7x sin(x)) as x approaches 0. This problem applies trigonometric approximations and limit properties to simplify and solve the expression. Perfect for calculus students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation