limit x->0 (e^x-1)/x
Let's evaluate the limit of <math><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo> -</mo><mn>1</mn></mrow><mrow><mi>x</mi></mrow></mfrac></math> when <m><math><mi>x</mi><mo> → </mo><mn>0</mn></math></m> The limit of <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math> is <m><math><mn>1</mn></math></m> when <m><math><mi>x</mi><mo> → </mo><mn>0</mn></math></m> The limit of <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo> -</mo><mn>1</mn></math> is <m><math><mn>0</mn></math></m> when <m><math><mi>x</mi><mo> → </mo><mn>0</mn></math></m> For <math><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo> -</mo><mn>1</mn></mrow><mrow><mi>x</mi></mrow></mfrac></math>, we encounter the <u>indeterminate form</u> <m>0/0</m> as <math><mi>x</mi></math> tends to <math><mn>0</mn></math> 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 differentiate: <math><mo>(</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo> -</mo><mn>1</mn><mo>)'</mo></math> <math><mo>(</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo>)'</mo><mo> + </mo><mn>0</mn></math> <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math> Let's perform the differentiation: <math><mo>(</mo><mi>x</mi><mo>)'</mo></math> <math><mn>1</mn></math> Using those derivatives, we obtain a new fraction whose limit can be computed: The limit of <math><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup></math> is <m><math><mn>1</mn></math></m> when <m><math><mi>x</mi><mo> → </mo><mn>0</mn></math></m> So the limit of <math><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>x</mi></mrow></msup><mo> -</mo><mn>1</mn></mrow><mrow><mi>x</mi></mrow></mfrac></math> is <m><math><mn>1</mn></math></m> when <m><math><mi>x</mi><mo> → </mo><mn>0</mn></math></m> ▷<b>Result: <math><mn>1</mn></math></b><span style='font-size:smaller'> (processing: 13 steps, 3 ms)</span>
