Question

Discuss the continuity of the function and evaluate the limit of f(x, y) (if it exists) as (x, y)\rightarrow (0, 0). f(x, y) = 1 - \frac{\cos(x^{2}+y^

Composite functions
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asked 2021-06-13

Discuss the continuity of the function and evaluate the limit of f(x, y) (if it exists) as \(\displaystyle{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}.{f{{\left({x},{y}\right)}}}={1}-{\frac{{{\cos{{\left({x}^{{{2}}}+{y}^{{{2}}}\right)}}}}}{{{x}^{{{2}}}+{y}^{{{2}}}}}}\)

Answers (1)

2021-06-14
The function is continuous everywhere except the origin (0,0) - since the cosine function is continuous everywhere and the denominator is 0 only at the origin.
the limit does not exist - one can consider \(\displaystyle{z}={x}^{{{2}}}+{y}^{{{2}}}\), when \(\displaystyle{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)},{z}\rightarrow{0}^{{+}}\), and \(\displaystyle{\cos{{\left({z}\right)}}}\rightarrow{1}\), but the denominator goes to \(\displaystyle\infty\), thus the limit does not exist.
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