Evaluatethe following limits:\operatorname{lit}_{x \rightarrow 0} \fra

1.	Evaluatethe following limits:\operatorname{lit}_{x \rightarrow 0} \frac{x \tan x}{1-\cos 2 x}
Answer» Lim(x→0) {xtanx}/(1 - cos2x ) we know, (1 - cos2x) =2sin²x , use this here, Lim(x→0){xtanx}/(2sin²x) =1/2 × Lim(x→0){ x ×( tanx/x)× x }/(sinx/x)² ×x² we know, Lim(f(x) →0) tanf(x)/f(x) = 1 Lim(f(x)→0) sinf(x)/f(x) = 1 use this concept now, = 1/2 × Lim(x→0) { x²/x²} = 1/2 × 1 = 1/2 ( answer )

Evaluatethe following limits:\operatorname{lit}_{x \rightarrow 0} \frac{x \tan x}{1-\cos 2 x}

Answer»

Lim(x→0) {xtanx}/(1 - cos2x )

we know, (1 - cos2x) =2sin²x , use this here,

Lim(x→0){xtanx}/(2sin²x)

=1/2 × Lim(x→0){ x ×( tanx/x)× x }/(sinx/x)² ×x²

we know, Lim(f(x) →0) tanf(x)/f(x) = 1 Lim(f(x)→0) sinf(x)/f(x) = 1

use this concept now,

= 1/2 × Lim(x→0) { x²/x²}

= 1/2 × 1

= 1/2 ( answer )