What is the limit as x approaches 0 of (3x^2)/(sqrt(1+xsin(x))-\sqrt{cos(x))} ?

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limit as x approaches 0 of (3x^2)/(sqrt(1+xsin(x))-\sqrt{cos(x))}

x \to 0 lim (\frac{3 x ^{2}}{1 + x sin ( x ) - cos ( x )})

4

Solution steps

x \to 0 lim (\frac{3 x ^{2}}{1 + x sin ( x ) - cos ( x )})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 3 \cdot x \to 0 lim (\frac{x ^{2}}{1 + x sin ( x ) - cos ( x )})

Apply L'Hopital's Rule

= 3 \cdot x \to 0 lim \frac{2 x}{\frac{x c o s ( x ) + s i n ( x )}{2 1 + x s i n ( x )} + \frac{s i n ( x )}{2 c o s ( x )}}

Apply L'Hopital's Rule

= 3 \cdot x \to 0 lim \frac{2}{\frac{- 2 x ^{2} s i n ^{2} ( x ) - x ^{2} c o s ^{2} ( x ) - 2 x s i n ( x ) + x s i n ( 2 x ) + 4 c o s ( x ) - s i n ^{2} ( x )}{4 ( 1 + x s i n ( x ) ) 1 + x s i n ( x )} + \frac{2 - s i n ^{2} ( x )}{4 c o s ( x ) c o s ( x )}}

Plug in the value

x = 0

= 3 \cdot \frac{2}{\frac{- 2 \cdot 0 ^{2} s i n ^{2} ( 0 ) - 0 ^{2} c o s ^{2} ( 0 ) - 2 \cdot 0 \cdot s i n ( 0 ) + 0 \cdot s i n ( 2 \cdot 0 ) + 4 c o s ( 0 ) - s i n ^{2} ( 0 )}{4 ( 1 + 0 \cdot s i n ( 0 ) ) 1 + 0 \cdot s i n ( 0 )} + \frac{2 - s i n ^{2} ( 0 )}{4 c o s ( 0 ) c o s ( 0 )}}

Simplify

3 \cdot \frac{2}{\frac{- 2 \cdot 0 ^{2} s i n ^{2} ( 0 ) - 0 ^{2} c o s ^{2} ( 0 ) - 2 \cdot 0 \cdot s i n ( 0 ) + 0 \cdot s i n ( 2 \cdot 0 ) + 4 c o s ( 0 ) - s i n ^{2} ( 0 )}{4 ( 1 + 0 \cdot s i n ( 0 ) ) 1 + 0 \cdot s i n ( 0 )} + \frac{2 - s i n ^{2} ( 0 )}{4 c o s ( 0 ) c o s ( 0 )}} : 4

= 4

What is the limit as x approaches 0 of (3x^2)/(sqrt(1+xsin(x))-\sqrt{cos(x))} ?
The limit as x approaches 0 of (3x^2)/(sqrt(1+xsin(x))-\sqrt{cos(x))} is 4