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

Here x → 0limsin ax/sin bx = x → 0lim [ sin ax/ax× ax ×1/sin bx/bx× bx] =x → 0lim [ sin ax/ax×1/sin bx/bx×ax/bx] =a/bx → 0lim [ sin ax/ax×1/sin bx/bx] = a/b× 1 ...

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Standard XI

Mathematics

L'Hospital Rule to Remove Indeterminate Form

Evaluate the ...

Question

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

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Solution

Here $lim x \to 0 \frac{sin ax}{sin bx}$

= $lim x \to 0 [\frac{sin ax}{ax} \times a x \times \frac{1}{\frac{sin bx}{b x} \times b x}]$

= $lim x \to 0 [\frac{sin ax}{ax} \times \frac{1}{\frac{s i n b x}{b x}} \times \frac{a x}{b x}]$

= $\frac{a}{b} lim x \to 0 [\frac{sin ax}{ax} \times \frac{1}{\frac{sin bx}{b x}}]$

= $\frac{a}{b} \times 1 \times 1 = \frac{a}{b}$

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Evaluate the following limit: limx→0sin axsin bx,a,b≠0

Here limx→0sin axsin bx = limx→0[sin axax×ax×1sin bxbx×bx] =limx→0[sin axax×1sin bxbx×axbx] =ablimx→0[sin axax×1sin bxbx] = ab×1×1=ab

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