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