Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of ((3+x)^2+(3-x)^2)/((3-x)^2-(3+x)^2)
-
Limit of (-1+x)/(-2+sqrt(3+x))
-
Limit of (-x+tan(x))/x^3
-
Limit of (-1+sqrt(x))/(-3+x)
- Graphing y =:
- tan(x)/x
- Derivative of:
-
tan(x)/x
- Integral of d{x}:
- tan(x)/x
-
Identical expressions
- tan(x)/x
- tangent of (x) divide by x
- tanx/x
- tan(x) divide by x
-
Similar expressions
- -sin(x)+(-x+tan(x))/x
- (-x+atan(x))/x^3
- (-x+tan(x))/x^3
- (-x+tan(x))/(x+2*sin(x))
Limit of the function tan(x)/x
The solution
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/tan(x)\ lim |------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right)$$
Limit(tan(x)/x, x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right)$$
transform
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right) = \lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) \lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}} = \lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
=
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
The limit
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
is first remarkable limit, is equal to 1.
The final answer:
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right)$$
transform
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right) = \lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) \lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}} = \lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
=
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
The limit
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
is first remarkable limit, is equal to 1.
The final answer:
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right) = 1$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \tan{\left(x \right)} = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(\tan^{2}{\left(x \right)} + 1\right)$$
=
$$\lim_{x \to 0^+}\left(\tan^{2}{\left(x \right)} + 1\right)$$
=
$$1$$
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
$$\lim_{x \to 0^+} \tan{\left(x \right)} = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(\tan^{2}{\left(x \right)} + 1\right)$$
=
$$\lim_{x \to 0^+}\left(\tan^{2}{\left(x \right)} + 1\right)$$
=
$$1$$
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]
/tan(x)\ lim |------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right)$$
1
$$1$$
= 1.0
/tan(x)\ lim |------| x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(x \right)}}{x}\right)$$
1
$$1$$
= 1.0
= 1.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(x \right)}}{x}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(x \right)}}{x}\right) = \tan{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(x \right)}}{x}\right) = \tan{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(x \right)}}{x}\right) = \tan{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(x \right)}}{x}\right) = \tan{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x}\right)$$
More at x→-oo
The graph