Question

Limit and Continuity Find the following of limit, if it exist. 

Answer 1

\( \lim\limits_{x \to 0} \frac{sin(3x²+x³)}{(3x²+x³}×\frac{3x²+x³}{6x²+x^7}(shape 0/0) =\lim\limits_{x \to 0} \frac{(3x²+x³)}{(6x²+x^7)} =\lim\limits_{x \to 0} \frac{(3+x)}{(6+x²)} =1/2 \)

Thus \( \lim\limits_{x \to 0}\frac{sin(3x²+x³)}{6x²+x^7} ) = \frac{1}{2} \)

\( \lim\limits_{x \to 0} \frac{sin(3x²+x³)}{(3x²+x³}×\frac{3x²+x³}{6x²+x^7}(shape 0/0) =\lim\limits_{x \to 0} \frac{(3x²+x³)}{(6x²+x^7)} =\lim\limits_{x \to 0} \frac{(3+x)}{(6+x²)} =1/2 \)

Thus \( \lim\limits_{x \to 0}\frac{sin(3x²+x³)}{6x²+x^7} ) = \frac{1}{2} \)

\( \lim\limits_{x \to 0} (sin(3x²+x³))/(6x²+7x^7) \) (Shape 0/0)