Evaluate the following : `lim_(x -> pi/4) [(sinx - cosx)^2/(sqrt(2) - sinx - cosx)]`

`lim_(x -> pi/4) [(sinx - cosx)^2/(sqrt(2) - sinx - cosx)]` = `lim_(x -> pi/4) (1 - sin2x)/(sqrt(2) - (sin x + cos x))` = `lim_(x -> pi/4) (1 - sin2x)/(sqrt(2) - sqrt(1 + sin2))` Put 1 + sin 2x = t &there4; sin 2x = t – 1 As `x -> pi/4, "t" -> 1 + sin 2 (pi/4)` &there4; `"t" -> 1 + sin pi/2` &there4; t → 1 + 1 &there4; t → 2 &there4; Required limit = `lim_("t" -> 2) (1 - ("t" - 1))/(sqrt(2) - sqrt("t"))` = `lim_("t" -> 2) (2 - "t")/(2^(1/2) - "t"^(1/2))` = `lim_("t" -> 2) ("t" - 2)/("t"^(1/2) - 2^(1/2))` = `lim_("t" -> 2) 1/(("t"^(1/2) - 2^(1/2))/("t" - 2)) ...[("Divide Numerator and Denominator"),("by" "t" - 2 "As" "t" -> 2 "," "t" ≠ 2),(therefore "t" - 2 ≠ 0)]` = `1/(1/2(2)^((-1)/2)` = `2(2)^(1/2)` = `2sqrt(2)`

Evaluate the following : limx→π4[(sinx-cosx)22-sinx-cosx] - Mathematics and Statistics

प्रश्न

Evaluate the following :

$lim_{x \to \frac{π}{4}} [\frac{{(\sin x - \cos x)}^{2}}{\sqrt{2} - \sin x - \cos x}]$

योग

उत्तर

$lim_{x \to \frac{π}{4}} [\frac{{(\sin x - \cos x)}^{2}}{\sqrt{2} - \sin x - \cos x}]$

= $lim_{x \to \frac{π}{4}} \frac{1 - \sin 2 x}{\sqrt{2} - (\sin x + \cos x)}$

= $lim_{x \to \frac{π}{4}} \frac{1 - \sin 2 x}{\sqrt{2} - \sqrt{1 + \sin 2}}$

Put 1 + sin 2x = t

∴ sin 2x = t – 1

As $x \to \frac{π}{4}, t \to 1 + \sin 2 (\frac{π}{4})$

∴ $t \to 1 + \sin \frac{π}{2}$

∴ t → 1 + 1

∴ t → 2

∴ Required limit

= $lim_{t \to 2} \frac{1 - (t - 1)}{\sqrt{2} - \sqrt{t}}$

= $lim_{t \to 2} \frac{2 - t}{2^{\frac{1}{2}} - t^{\frac{1}{2}}}$

= $lim_{t \to 2} \frac{t - 2}{t^{\frac{1}{2}} - 2^{\frac{1}{2}}}$

= $lim_{t \to 2} \frac{1}{\frac{t^{\frac{1}{2}} - 2^{\frac{1}{2}}}{t - 2}} ... [\begin{matrix} Divide Numerator and Denominator \\ by t - 2 As t \to 2, t \neq 2 \\ ∴ t - 2 \neq 0 \end{matrix}]$

= $\frac{1}{\frac{1}{2} {(2)}^{\frac{- 1}{2}}}$

= $2 {(2)}^{\frac{1}{2}}$

= $2 \sqrt{2}$

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Limits of Trigonometric Functions

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q II. (17)Q II. (16)Q II. (18)

अध्याय 7: Limits - Miscellaneous Exercise 7.2 [पृष्ठ १५९]

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बालभारती Mathematics and Statistics 2 (Arts and Science) [English] 11 Standard Maharashtra State Board

अध्याय 7 Limits
Miscellaneous Exercise 7.2 | Q II. (17) | पृष्ठ १५९

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Evaluate the following : limx→π4[(sinx-cosx)22-sinx-cosx] - Mathematics and Statistics

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