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

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