The value of \[\mathop {\lim }\limits_{{\rm{x}} \to 0} \frac{{1 – \cos \left( {{{\rm{x}}^2}} \right)}}{{2{x^4}}}\] is A. 0 B. \[\frac{1}{2}\] C. \[\frac{1}{4}\] D. undefined

” option3=”\[\frac{1}{4}\]” option4=”undefined” correct=”option2″]

The correct answer is $\boxed{\text{D. undefined}}$.

To see why, let’s first consider the limit as $x$ approaches 0 from the positive side. As $x$ gets closer and closer to 0, the cosine function approaches 1. This means that the numerator, $1 – \cos(x^2)$, approaches 0. The denominator, $2x^4$, also approaches 0, but much more quickly. This is because $x^4$ is a much larger number than $x^2$ when $x$ is close to 0. So, the limit as $x$ approaches 0 from the positive side is 0 divided by a very small number, which is undefined.

Now, let’s consider the limit as $x$ approaches 0 from the negative side. As $x$ gets closer and closer to 0 from the negative side, the cosine function approaches -1. This means that the numerator, $1 – \cos(x^2)$, approaches -2. The denominator, $2x^4$, is still approaching 0, but it is still approaching 0 from the positive side. So, the limit as $x$ approaches 0 from the negative side is -2 divided by a very small number, which is also undefined.

Therefore, the limit as $x$ approaches 0 does not exist, and the answer is $\boxed{\text{D. undefined}}$.