Solve the previous limit without using l'hôpital's Rule 

To solve the limit without using **L'Hôpital's Rule**, we can utilize a **trigonometric identity** and make some algebraic manipulations.

The limit is:

\[
\lim_{x \to \frac{\pi}{4}} \frac{\tan x - 1}{x - \frac{\pi}{4}}
\]

### Step 1: Substitution
Let \( h = x - \frac{\pi}{4} \), so as \( x \to \frac{\pi}{4} \), \( h \to 0 \). Then \( x = h + \frac{\pi}{4} \). Substitute this into the expression:

\[
\lim_{h \to 0} \frac{\tan\left(h + \frac{\pi}{4}\right) - 1}{h}
\]

### Step 2: Use the Sum Identity for Tangent
The tangent sum identity is:
\[
\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \cdot \tan b}
\]
Let \( a = h \) and \( b = \frac{\pi}{4} \). Since \( \tan\left(\frac{\pi}{4}\right) = 1 \), the identity simplifies to:
\[
\tan\left(h + \frac{\pi}{4}\right) = \frac{\tan h + 1}{1 - \tan h}
\]

Substituting this into our limit:
\[
\lim_{h \to 0} \frac{\frac{\tan h + 1}{1 - \tan h} - 1}{h}
\]

### Step 3: Simplify the Expression
Now simplify the numerator:
\[
\frac{\tan h + 1}{1 - \tan h} - 1 = \frac{\tan h + 1 - (1 - \tan h)}{1 - \tan h} = \frac{2\tan h}{1 - \tan h}
\]

Thus, the limit becomes:
\[
\lim_{h \to 0} \frac{\frac{2\tan h}{1 - \tan h}}{h} = \lim_{h \to 0} \frac{2\tan h}{h(1 - \tan h)}
\]

### Step 4: Evaluate the Limit
As \( h \to 0 \), \( \tan h \approx h \) (since \( \tan h \) is approximately equal to \( h \) for small angles). So:
\[
\frac{2\tan h}{h(1 - \tan h)} \approx \frac{2h}{h(1 - h)} = \frac{2}{1} = 2
\]

Thus, the value of the limit is:
\[
\boxed{2}
\]

### Conclusion
We have solved the limit using trigonometric identities and algebraic manipulations without L'Hôpital's Rule. The final result remains \( 2 \).

Would you like to explore any of the steps in more detail?

Here are five additional questions that could extend your understanding:
1. What other trigonometric identities are useful in limit problems?
2. How do Taylor series approximations help in limit evaluation?
3. What are the common small-angle approximations for trigonometric functions?
4. How can substitution techniques simplify complex limits?
5. How can we approach limits involving multiple trigonometric functions?

**Tip**: When solving limits involving trigonometric functions, small-angle approximations like \( \sin x \approx x \) and \( \tan x \approx x \) near zero can be very useful!

Learn how to solve the limit \( \lim_{x \to \frac{\pi}{4}} \frac{\tan x - 1}{x - \frac{\pi}{4}} \) without using L'Hôpital's Rule. Explore step-by-step trigonometric identities and algebraic manipulations to find the solution, achieving a final result of 2. Suitable for advanced high school students and above.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation