Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of exp(x)/x
-
Limit of tan(4*x)
-
Limit of sqrt(3)/2
- Limit of x^2*y^2
- Integral of d{x}:
-
exp(x)/x
- Graphing y =:
- exp(x)/x
-
Identical expressions
- exp(x)/x
- exponent of (x) divide by x
- expx/x
- exp(x) divide by x
-
Similar expressions
- (-exp(-x)-2*x+exp(x))/(x-sin(x))
- (-cos(x)+exp(x))/x
- (-exp(-x)+exp(x))/x
Limit of the function exp(x)/x
The solution
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/ x\
|e |
lim |--|
x->oo\x /
$$\lim_{x \to \infty}\left(\frac{e^{x}}{x}\right)$$
Limit(exp(x)/x, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} e^{x} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{e^{x}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} e^{x}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty} e^{x}$$
=
$$\lim_{x \to \infty} e^{x}$$
=
$$\infty$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty} e^{x} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{e^{x}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} e^{x}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty} e^{x}$$
=
$$\lim_{x \to \infty} e^{x}$$
=
$$\infty$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{e^{x}}{x}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{e^{x}}{x}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{x}}{x}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{e^{x}}{x}\right) = e$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{x}}{x}\right) = e$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{x}}{x}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{e^{x}}{x}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{x}}{x}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{e^{x}}{x}\right) = e$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{x}}{x}\right) = e$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{x}}{x}\right) = 0$$
More at x→-oo
The graph