Mister Exam

Lang:

How to use it?
Limit of the function:
Limit of (1+n)/(2+n)
Limit of (1-7/x)^x
Limit of ((-2+x)/(1+x))^(-3+2*x)
Limit of (-tan(a)+tan(x))/(x-a)
Identical expressions
tan(three *x)/(two *sin(five *x))
tangent of (3 multiply by x) divide by (2 multiply by sinus of (5 multiply by x))
tangent of (three multiply by x) divide by (two multiply by sinus of (five multiply by x))
tan(3x)/(2sin(5x))
tan3x/2sin5x
tan(3*x) divide by (2*sin(5*x))

Limit of the function tan(3x)/(2sin(5*x))

You have entered [src]

     / tan(3*x) \
 lim |----------|
x->0+\2*sin(5*x)/

\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{2 \sin{\left(5 x \right)}}\right)

Limit(tan(3*x)/((2*sin(5*x))), x, 0)

Lopital's rule

We have indeterminateness of type

0/0,

i.e. limit for the numerator is

\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{2}\right) = 0

and limit for the denominator is

\lim_{x \to 0^+} \sin{\left(5 x \right)} = 0

Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.

\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{2 \sin{\left(5 x \right)}}\right)

=
Let's transform the function under the limit a few

\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{2 \sin{\left(5 x \right)}}\right)

\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\tan{\left(3 x \right)}}{2}}{\frac{d}{d x} \sin{\left(5 x \right)}}\right)

\lim_{x \to 0^+}\left(\frac{\frac{3 \tan^{2}{\left(3 x \right)}}{2} + \frac{3}{2}}{5 \cos{\left(5 x \right)}}\right)

\lim_{x \to 0^+}\left(\frac{3 \tan^{2}{\left(3 x \right)}}{10} + \frac{3}{10}\right)

\lim_{x \to 0^+}\left(\frac{3 \tan^{2}{\left(3 x \right)}}{10} + \frac{3}{10}\right)

\frac{3}{10}

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

Plot the graph

Other limits x→0, -oo, +oo, 1

\lim_{x \to 0^-}\left(\frac{\tan{\left(3 x \right)}}{2 \sin{\left(5 x \right)}}\right) = \frac{3}{10}

\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{2 \sin{\left(5 x \right)}}\right) = \frac{3}{10}

\lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)}}{2 \sin{\left(5 x \right)}}\right)

\lim_{x \to 1^-}\left(\frac{\tan{\left(3 x \right)}}{2 \sin{\left(5 x \right)}}\right) = \frac{\tan{\left(3 \right)}}{2 \sin{\left(5 \right)}}

\lim_{x \to 1^+}\left(\frac{\tan{\left(3 x \right)}}{2 \sin{\left(5 x \right)}}\right) = \frac{\tan{\left(3 \right)}}{2 \sin{\left(5 \right)}}

\lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)}}{2 \sin{\left(5 x \right)}}\right)

One‐sided limits [src]

     / tan(3*x) \
 lim |----------|
x->0+\2*sin(5*x)/

\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{2 \sin{\left(5 x \right)}}\right)

3/10

\frac{3}{10}

= 0.3

     / tan(3*x) \
 lim |----------|
x->0-\2*sin(5*x)/

\lim_{x \to 0^-}\left(\frac{\tan{\left(3 x \right)}}{2 \sin{\left(5 x \right)}}\right)

3/10

\frac{3}{10}

= 0.3

= 0.3

Rapid solution [src]

3/10

\frac{3}{10}

Expand and simplify

Numerical answer [src]

0.3

0.3

The graph

Limit of the function tan(3*x)/(2*sin(5*x))