The value of lim x → 0 ⁡ 1 + tan ⁡ x 1 + sin ⁡ x 2 sin ⁡ x is equal to

Step-by-step solution

Let, f x = 1 + tan ⁡ x 1 + sin ⁡ x and g x = 2 sin ⁡ x Clearly, f x → 1 and g x → ∞ as x → 0 . ∴ lim x → 0 ⁡ 1 + tan ⁡ x 1 + sin ⁡ x 2 sin ⁡ x = e lim x → 0 2 sin ⁡ x ⁡ 1 + tan ⁡ x 1 + sin ⁡ x - 1 using   lim x → a ⁡ f x g x = e lim x → a g x ⁡ f x - 1 for   1 ∞   form = e lim x → 0 ⁡ 2 sin ⁡ x tan ⁡ x - sin ⁡ x 1 + sin ⁡ x = e l