Question

Find the limit (if it exists) and discuss the continuity of the function. \lim_{(x, y)→(0, 0)} \frac{y+xe^{-y²}}{1+x²}

Limits and continuity
ANSWERED
asked 2021-05-16
Find the limit (if it exists) and discuss the continuity of the function. \(\displaystyle\lim_{{{\left({x},{y}\right)}→{\left({0},{0}\right)}}}{\frac{{{y}+{x}{e}^{{-{y}²}}}}{{{1}+{x}²}}}\)

Expert Answers (1)

2021-05-17
The continuity and limit of the two variable function
\(\displaystyle\lim_{{{\left({x},{y}\right)}→{\left({0},{0}\right)}}}{\frac{{{y}+{x}{e}^{{-{y}²}}}}{{{1}+{x}²}}}\)
According to the continuity equation of two variable have to follow
\(\displaystyle\lim_{{{\left({x},{y}\right)}→{\left({x}_{{{0}}},{y}_{{{0}}}\right)}}}{f{{\left({x},{y}\right)}}}={f{{\left({x}_{{{0}}},{y}_{{{0}}}\right)}}}\) Here we have to check whether it is follow or not
\(\displaystyle\lim_{{{\left({x},{y}\right)}→{\left({0},{0}\right)}}}{\frac{{{y}+{x}{e}^{{-{y}²}}}}{{{1}+{x}²}}}={0}{\quad\text{and}\quad}{f{{\left({0},{0}\right)}}}={0}\)
Thus the given function is continuous in the open region R which means there is no point of discontinuous and limit of function
\(\displaystyle\lim_{{{\left({x},{y}\right)}→{\left({0},{0}\right)}}}{\frac{{{y}+{x}{e}^{{-{y}²}}}}{{{1}+{x}²}}}={0}\)
exist that is 0
0
 
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