\( \lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { \cos x - 1 - x ^ { 2 } / 2 } { x ^ { 4 } + y ^ { 4 } } \)

To evaluate the limit \[ \lim_{(x, y) \rightarrow (0, 0)} \frac{\cos x - 1 - \frac{x^2}{2}}{x^4 + y^4}, \] we start by simplifying the numerator. We can use the Taylor series expansion of \(\cos x\) around \(x = 0\): \[ \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + O(x^8). \] Subtracting \(1\) and \(\frac{x^2}{2}\) from \(\cos x\), we have: \[ \cos x - 1 - \frac{x^2}{2} = \left(1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + O(x^8)\right) - 1 - \frac{x^2}{2} = \frac{x^4}{24} - \frac{x^6}{720} + O(x^8). \] Thus, the numerator simplifies to: \[ \cos x - 1 - \frac{x^2}{2} \approx \frac{x^4}{24} \quad \text{as } (x, y) \to (0, 0). \] Now, substituting this back into our limit, we get: \[ \lim_{(x, y) \rightarrow (0, 0)} \frac{\frac{x^4}{24} + O(x^6)}{x^4 + y^4}. \] As \((x, y) \to (0, 0)\), the dominant term in the numerator is \(\frac{x^4}{24}\). Therefore, we can focus on the leading term: \[ \lim_{(x, y) \rightarrow (0, 0)} \frac{\frac{x^4}{24}}{x^4 + y^4}. \] Next, we analyze the denominator \(x^4 + y^4\). We can factor out \(x^4\) when \(x\) is not zero: \[ x^4 + y^4 = x^4(1 + \left(\frac{y}{x}\right)^4). \] Thus, we rewrite the limit as: \[ \lim_{(x, y) \rightarrow (0, 0)} \frac{\frac{x^4}{24}}{x^4(1 + \left(\frac{y}{x}\right)^4)} = \lim_{(x, y) \rightarrow (0, 0)} \frac{1}{24(1 + \left(\frac{y}{x}\right)^4)}. \] Now, we need to consider the behavior of \(\frac{y}{x}\) as \((x, y) \to (0, 0)\). If we approach along the line \(y = kx\) for some constant \(k\), we have: \[ \frac{y}{x} = k \implies \left(\frac{y}{x}\right)^4 = k^4. \] Thus, the limit becomes: \[ \frac{1}{24(1 + k^4)}. \] Since \(k\) can take any value, the limit depends on the path taken to approach \((0, 0)\). Therefore, the limit does not exist because it varies with different values of \(k\). In conclusion, the limit is: \[ \boxed{\text{does not exist}}. \]