Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-1+e^x)/sin(x)
-
Limit of ((-1+x)/(4+x))^(2+3*x)
-
Limit of (1+n)/(2+n)
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Limit of (3+2*x)/(1+5*x)
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Identical expressions
- (- one +e^x)/sin(x)
- ( minus 1 plus e to the power of x) divide by sinus of (x)
- ( minus one plus e to the power of x) divide by sinus of (x)
- (-1+ex)/sin(x)
- -1+ex/sinx
- -1+e^x/sinx
- (-1+e^x) divide by sin(x)
-
Similar expressions
- (1+e^x)/sin(x)
- (-1-e^x)/sin(x)
- (-1+e^x)/sinx
Limit of the function (-1+e^x)/sin(x)
The solution
You have entered
[src]
/ x\
|-1 + E |
lim |-------|
x->0+\ sin(x)/
$$\lim_{x \to 0^+}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right)$$
Limit((-1 + E^x)/sin(x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(e^{x} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{x} - 1\right)}{\frac{d}{d x} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{e^{x}}{\cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{e^{x}}{\cos{\left(x \right)}}\right)$$
=
$$1$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(e^{x} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{x} - 1\right)}{\frac{d}{d x} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{e^{x}}{\cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{e^{x}}{\cos{\left(x \right)}}\right)$$
=
$$1$$
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
One‐sided limits
[src]
/ x\
|-1 + E |
lim |-------|
x->0+\ sin(x)/
$$\lim_{x \to 0^+}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right)$$
1
$$1$$
= 1.0
/ x\
|-1 + E |
lim |-------|
x->0-\ sin(x)/
$$\lim_{x \to 0^-}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right)$$
1
$$1$$
= 1.0
= 1.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right) = \frac{-1 + e}{\sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right) = \frac{-1 + e}{\sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right) = \frac{-1 + e}{\sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right) = \frac{-1 + e}{\sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{e^{x} - 1}{\sin{\left(x \right)}}\right)$$
More at x→-oo
The graph