X-limxcos -4x sin -4x - -b- Find b

Substitute #160; ( π /4x )= y ∴ y→ 0 when x→∞ Limit = y→ 0limπ4ycos ysin y= y→ 0limπ4.cos y.sin yy = π4.1.= π4. ⇒ b=4 ...

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Substitution Method to Remove Indeterminate Form

x→∞limxcos π/...

Question

$l i m x \to \infty x cos (\frac{π}{4 x}) sin (\frac{π}{4 x}) = \frac{π}{b} .$ Find b

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lim x \to \infty x c o s (π / 4 x) s i n (π / 4 x)

f (x) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} \frac{1 - \sqrt{2} sin x}{π - 4 x}, i f x \neq \frac{π}{4} b, i f x = \frac{π}{4} \end{matrix}

x = \frac{π}{4}

b

f (x) = \{\begin{matrix} \frac{x - 4}{|x - 4|} + a, if & x < 4 \\ a + b, if & x = 4 \\ \frac{x - 4}{|x - 4|} + b, if & x > 4 \end{matrix}

f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} x + a \sqrt{2} sin x, & 0 \leq x < π / 4 2 x cot x + b, & π / 4 \leq x \leq π / 2 a cos 2 x - b sin x, & π / 2 < x \leq π \end{matrix}

0 \leq x \leq π .

(x + a) (x + b) = x^{2} + (a + b) x + a b

(4 x - 5) (4 x - 1)

(4 x + 5) (4 x - 1)

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