1) Let's evaluate the limit of <math><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow></mfrac></math> when <m><math><mi>x</mi><mo> → </mo><mi>+∞</mi></math></m>
The limit of <math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></math> is <m><math><mi>+∞</mi></math></m> when <m><math><mi>x</mi><mo> → </mo><mi>+∞</mi></math></m>
The limit of <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math> is <m><math><mi>+∞</mi></math></m> when <m><math><mi>x</mi><mo> → </mo><mi>+∞</mi></math></m>
For <math><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow></mfrac></math>, we encounter the <u>indeterminate form</u> <math><mfrac><mrow><mi>+∞</mi></mrow><mrow><mi>+∞</mi></mrow></mfrac></math> as <math><mi>x</mi></math> tends to <math><mi>+∞</mi></math>
2) We can use L'Hôpital's rule (<m>limif of a/b = limit of a'/b'</m>) by computing the derivatives of the numerator and the denominator:
Let's perform the differentiation: <math><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)'</mo></math>
 <math><mn>2</mn><mi>x</mi></math>
3) Let's perform the differentiation: <math><mo>(</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo>)'</mo></math>
 <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math>
With the derivatives of the numerator and the denominator, we obtain a new fraction whose limit can be determined:
 The limit of <math><mn>2</mn><mi>x</mi></math> is <m><math><mi>+∞</mi></math></m> when <m><math><mi>x</mi><mo> → </mo><mi>+∞</mi></math></m>
 The limit of <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math> is <m><math><mi>+∞</mi></math></m> when <m><math><mi>x</mi><mo> → </mo><mi>+∞</mi></math></m>
For <math><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow></mfrac></math>, we encounter the <u>indeterminate form</u> <math><mfrac><mrow><mi>+∞</mi></mrow><mrow><mi>+∞</mi></mrow></mfrac></math> as <math><mi>x</mi></math> tends to <math><mi>+∞</mi></math>
For <math><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow></mfrac></math>, the limit takes the <u>indeterminate form</u> <math><mfrac><mrow><mi>+∞</mi></mrow><mrow><mi>+∞</mi></mrow></mfrac></math> when <math><mi>x</mi></math> → <math><mi>+∞</mi></math>
4) We can use L'Hôpital's rule (<m>limif of a/b = limit of a'/b'</m>) by computing the derivatives of the numerator and the denominator:
Let's perform the differentiation: <math><mo>(</mo><mn>2</mn><mi>x</mi><mo>)'</mo></math>
 <math><mn>2</mn></math>
5) Let's perform the differentiation: <math><mo>(</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo>)'</mo></math>
 <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math>
With the derivatives of the numerator and the denominator, we obtain a new fraction whose limit can be determined:
 The limit of <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math> is <m><math><mi>+∞</mi></math></m> when <m><math><mi>x</mi><mo> → </mo><mi>+∞</mi></math></m>
 The limit of <math><mfrac><mrow><mn>2</mn></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow></mfrac></math> is <m><math><mn>0</mn></math>⁺</m> when <m><math><mi>x</mi><mo> → </mo><mi>+∞</mi></math></m>
The limit of <math><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow></mfrac></math> is <m><math><mn>0</mn></math>⁺</m> when <m><math><mi>x</mi><mo> → </mo><mi>+∞</mi></math></m>
So the limit of <math><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow></mfrac></math> is <m><math><mn>0</mn></math>⁺</m> when <m><math><mi>x</mi><mo> → </mo><mi>+∞</mi></math></m>

▷<b>Result: &nbsp; <math><mn>0</mn></math></b><span style='font-size:smaller'> &nbsp; &nbsp; (computation required 24 steps and 3 ms)</span>