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- Homework Statement:
- Find the limit of the function below as ##\textbf{x}\to\textbf{0}##.

- Relevant Equations:
- ##\frac{e^{|\textbf{x}|^2}-1}{|\textbf{x}|^2+x^2_1x_2+x^2_2x_3}## where ##\textbf{x}=(x_1,x_2,x_3)## and ##|\textbf{x}|=\sqrt{x^2_1+x^2_2+x^2_3}##.

If one approaches the origin from where ##x_2=0##, the terms ##x^2_1x_2+x^2_2x_3## in the denominator equal ##0##. Substituting ##|\textbf{x}|^2## for ##t## yields the expression ##\frac{e^t-1}{t}##, which has limit 1 as ##\textbf{x}\to\textbf{0}## and thus ##t\to0##. So the limit should be 1 if it exists. How could one show that it does exist?