Evaluate the following limit-lim x - 0sin a x-b x-a x-sin b x- a- b- a-b-0

Here x→ 0limsin ax+bx/ax+sin bx [ 0/0form] Dividing numerator and denominator by ax, x→ 0limsin ax/ax+bx/ax/ax/ax+sin bx/ax = x→ 0limsin ax/ax+bx/ax/ax/ax+sin ...

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Standard XI

Mathematics

General Solutions of (sin theta)^2 = (sin alpha)^2 , (cos theta)^2 = (cos alpha)^2 , (tan theta)^2 = (tan alpha)^2

Evaluate the ...

Question

Evaluate the following limit:
$lim x \to 0 \frac{sin ax + bx}{ax+sin bx}, a, b, a + b \neq 0$

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Solution

Here $lim x \to 0 \frac{sin ax+bx}{ax+sin bx} [\frac{0}{0} form]$

Dividing numerator and denominator by ax,

$lim x \to 0 \frac{\frac{sin ax}{ax} + \frac{b x}{a x}}{\frac{a x}{a x} + \frac{sin bx}{a x}} = lim x \to 0 \frac{\frac{sin ax}{ax} + \frac{b x}{a x}}{\frac{a x}{a x} + \frac{sin bx}{b x} \times \frac{b x}{a x}}$

= $\frac{1 + \frac{b}{a}}{1 + \frac{b}{a}}$ = 1.

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