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To find the limit of the expression [tex]\( \lim _{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2} - 2 \cos x}{4x - \pi} \)[/tex], we will follow a systematic approach. Here's the detailed step-by-step solution:
1. Identify the Limit Form:
First, identify the form of the limit when [tex]\( x \)[/tex] approaches [tex]\( \frac{\pi}{4} \)[/tex].
- Substituting [tex]\( x = \frac{\pi}{4} \)[/tex] into the numerator [tex]\( \sqrt{2} - 2 \cos x \)[/tex] results in:
[tex]\[ \sqrt{2} - 2 \cos \left( \frac{\pi}{4} \right) = \sqrt{2} - 2 \left( \frac{\sqrt{2}}{2} \right) = \sqrt{2} - \sqrt{2} = 0 \][/tex]
- Substituting [tex]\( x = \frac{\pi}{4} \)[/tex] into the denominator [tex]\( 4x - \pi \)[/tex] results in:
[tex]\[ 4 \left( \frac{\pi}{4} \right) - \pi = \pi - \pi = 0 \][/tex]
Since both the numerator and the denominator approach 0 as [tex]\( x \)[/tex] approaches [tex]\( \frac{\pi}{4} \)[/tex], we have an indeterminate form [tex]\( \frac{0}{0} \)[/tex].
2. Apply L'Hôpital's Rule:
Because we have an indeterminate form [tex]\( \frac{0}{0} \)[/tex], we can apply L'Hôpital's Rule, which states that:
[tex]\[ \lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)} \text{ if }\lim _{x \rightarrow c} \frac{f(x)}{g(x)} \text{ is indeterminate form of } \frac{0}{0} \text{ or } \frac{\infty}{\infty} \][/tex]
3. Differentiate the Numerator and the Denominator:
- Differentiate the numerator [tex]\( \sqrt{2} - 2 \cos x \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{d}{dx} (\sqrt{2} - 2 \cos x) = 0 + 2 \sin x = 2 \sin x \][/tex]
- Differentiate the denominator [tex]\( 4x - \pi \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{d}{dx} (4x - \pi) = 4 \][/tex]
4. Substitute the Limits in the Derivatives:
Now, we substitute [tex]\( x = \frac{\pi}{4} \)[/tex] into the derivatives:
[tex]\[ \lim_{x \to \frac{\pi}{4}} \frac{2 \sin x}{4} = \frac{2 \sin \left( \frac{\pi}{4} \right)}{4} \][/tex]
5. Simplify the Expression:
- Evaluate [tex]\( \sin \left( \frac{\pi}{4} \right) \)[/tex]:
[tex]\[ \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \][/tex]
- Substitute back:
[tex]\[ \frac{2 \cdot \frac{\sqrt{2}}{2}}{4} = \frac{\sqrt{2}}{4} \][/tex]
Thus, the limit is:
[tex]\[ \lim _{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2} - 2 \cos x}{4x - \pi} = \frac{\sqrt{2}}{4} \][/tex]
1. Identify the Limit Form:
First, identify the form of the limit when [tex]\( x \)[/tex] approaches [tex]\( \frac{\pi}{4} \)[/tex].
- Substituting [tex]\( x = \frac{\pi}{4} \)[/tex] into the numerator [tex]\( \sqrt{2} - 2 \cos x \)[/tex] results in:
[tex]\[ \sqrt{2} - 2 \cos \left( \frac{\pi}{4} \right) = \sqrt{2} - 2 \left( \frac{\sqrt{2}}{2} \right) = \sqrt{2} - \sqrt{2} = 0 \][/tex]
- Substituting [tex]\( x = \frac{\pi}{4} \)[/tex] into the denominator [tex]\( 4x - \pi \)[/tex] results in:
[tex]\[ 4 \left( \frac{\pi}{4} \right) - \pi = \pi - \pi = 0 \][/tex]
Since both the numerator and the denominator approach 0 as [tex]\( x \)[/tex] approaches [tex]\( \frac{\pi}{4} \)[/tex], we have an indeterminate form [tex]\( \frac{0}{0} \)[/tex].
2. Apply L'Hôpital's Rule:
Because we have an indeterminate form [tex]\( \frac{0}{0} \)[/tex], we can apply L'Hôpital's Rule, which states that:
[tex]\[ \lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)} \text{ if }\lim _{x \rightarrow c} \frac{f(x)}{g(x)} \text{ is indeterminate form of } \frac{0}{0} \text{ or } \frac{\infty}{\infty} \][/tex]
3. Differentiate the Numerator and the Denominator:
- Differentiate the numerator [tex]\( \sqrt{2} - 2 \cos x \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{d}{dx} (\sqrt{2} - 2 \cos x) = 0 + 2 \sin x = 2 \sin x \][/tex]
- Differentiate the denominator [tex]\( 4x - \pi \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{d}{dx} (4x - \pi) = 4 \][/tex]
4. Substitute the Limits in the Derivatives:
Now, we substitute [tex]\( x = \frac{\pi}{4} \)[/tex] into the derivatives:
[tex]\[ \lim_{x \to \frac{\pi}{4}} \frac{2 \sin x}{4} = \frac{2 \sin \left( \frac{\pi}{4} \right)}{4} \][/tex]
5. Simplify the Expression:
- Evaluate [tex]\( \sin \left( \frac{\pi}{4} \right) \)[/tex]:
[tex]\[ \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \][/tex]
- Substitute back:
[tex]\[ \frac{2 \cdot \frac{\sqrt{2}}{2}}{4} = \frac{\sqrt{2}}{4} \][/tex]
Thus, the limit is:
[tex]\[ \lim _{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2} - 2 \cos x}{4x - \pi} = \frac{\sqrt{2}}{4} \][/tex]
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