Answer
(a) We can see that $\lim\limits_{x \to 0} (1+x)^{1/x} = 2.71828 = e$
(b) On the graph, when the function $f(x) = (1+x)^{1/x}$ crosses the y-axis, we can see that $\lim\limits_{x \to 0} (1+x)^{1/x} = e$
Work Step by Step
(a) We can evaluate $f(x) = (1+x)^{1/x}$ for values of $x$ that approach $0$:
$f(0.1) = (1+0.1)^{1/0.1} = 2.59374$
$f(0.01) = (1+0.01)^{1/0.01} = 2.70481$
$f(0.001) = (1+0.001)^{1/0.001} = 2.71692$
$f(0.0001) = (1+0.0001)^{1/0.0001} = 2.71815$
$f(0.00001) = (1+0.0001)^{1/0.00001} = 2.71827$
$f(0.000001) = (1+0.00001)^{1/0.000001} = 2.71828$
$f(0.0000001) = (1+0.000001)^{1/0.0000001} = 2.71828$
$f(-0.1) = (1-0.1)^{1/-0.1} = 2.86797$
$f(-0.01) = (1-0.01)^{1/-0.01} = 2.73200$
$f(-0.001) = (1-0.001)^{1/-0.001} = 2.71964$
$f(-0.0001) = (1-0.0001)^{1/-0.0001} = 2.71842$
$f(-0.00001) = (1-0.0001)^{1/-0.00001} = 2.71830$
$f(-0.000001) = (1-0.00001)^{1/-0.000001} = 2.71828$
$f(-0.0000001) = (1-0.000001)^{1/-0.0000001} = 2.71828$
We can see that $\lim\limits_{x \to 0} (1+x)^{1/x} = 2.71828 = e$
(b) On the graph, when the function $f(x) = (1+x)^{1/x}$ crosses the y-axis, we can see that $\lim\limits_{x \to 0} (1+x)^{1/x} = e$
