Math Problem Statement

lim (x → π/4) (1 - tan(x)) / (sin(x) - cos(x))

Solution

We are tasked with evaluating the limit:

limxπ41tan(x)sin(x)cos(x)\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan(x)}{\sin(x) - \cos(x)}

Step 1: Substitute x=π4x = \frac{\pi}{4}

First, let’s substitute x=π4x = \frac{\pi}{4} directly into the expression:

  • tan(π4)=1\tan\left(\frac{\pi}{4}\right) = 1
  • sin(π4)=cos(π4)=22\sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}

So, at x=π4x = \frac{\pi}{4}, the expression becomes:

112222=00\frac{1 - 1}{\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}} = \frac{0}{0}

This results in an indeterminate form 00\frac{0}{0}, so we need to apply L'Hopital's Rule, which allows us to differentiate the numerator and denominator separately and then evaluate the limit.

Step 2: Apply L'Hopital's Rule

We need to differentiate the numerator and the denominator:

  • The numerator is 1tan(x)1 - \tan(x). The derivative of 11 is 0, and the derivative of tan(x)-\tan(x) is sec2(x)-\sec^2(x). So, the derivative of the numerator is:

    sec2(x)-\sec^2(x)

  • The denominator is sin(x)cos(x)\sin(x) - \cos(x). The derivative of sin(x)\sin(x) is cos(x)\cos(x), and the derivative of cos(x)-\cos(x) is sin(x)\sin(x). So, the derivative of the denominator is:

    cos(x)+sin(x)\cos(x) + \sin(x)

Step 3: Evaluate the new limit

Now we evaluate the new limit:

limxπ4sec2(x)cos(x)+sin(x)\lim_{x \to \frac{\pi}{4}} \frac{-\sec^2(x)}{\cos(x) + \sin(x)}

  • sec2(π4)=1cos2(π4)=1(22)2=2\sec^2\left(\frac{\pi}{4}\right) = \frac{1}{\cos^2\left(\frac{\pi}{4}\right)} = \frac{1}{\left(\frac{\sqrt{2}}{2}\right)^2} = 2
  • cos(π4)+sin(π4)=22+22=2\cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2}

So, the limit becomes:

22=2\frac{-2}{\sqrt{2}} = -\sqrt{2}

Final Answer:

limxπ41tan(x)sin(x)cos(x)=2\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan(x)}{\sin(x) - \cos(x)} = -\sqrt{2}

Would you like more details on any step? Here are some related questions you might find useful:

  1. How do you apply L'Hopital's Rule in more complicated cases?
  2. What are the conditions under which L'Hopital’s Rule can be used?
  3. Can limits be solved using series expansions instead of L'Hopital’s Rule?
  4. How do you calculate derivatives of trigonometric functions like sec2(x)\sec^2(x)?
  5. What is the geometric interpretation of the limit problem?

Tip: Always check the form of your limit before applying L'Hopital’s Rule to ensure it’s indeterminate, such as 00\frac{0}{0} or \frac{\infty}{\infty}.

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Math Problem Analysis

Mathematical Concepts

Limits
Trigonometry
L'Hopital's Rule
Derivatives

Formulas

lim (x → c) f(x) / g(x) = lim (x → c) f'(x) / g'(x) (L'Hopital's Rule)
Derivative of tan(x) is sec^2(x)
sin(π/4) = cos(π/4) = √2/2
sec^2(π/4) = 2

Theorems

L'Hopital's Rule

Suitable Grade Level

College-level Calculus