Mister exam

Limit of the function cot(pi*x)*sin(x)/(2*sec(x))

at

from to

The solution

You have entered [src]
     /cot(pi*x)*sin(x)\
lim |----------------|
x->0+\    2*sec(x)    /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cot{\left(\pi x \right)}}{2 \sec{\left(x \right)}}\right)$$
Limit((cot(pi*x)*sin(x))/((2*sec(x))), x, 0)
The graph
One‐sided limits [src]
     /cot(pi*x)*sin(x)\
lim |----------------|
x->0+\    2*sec(x)    /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cot{\left(\pi x \right)}}{2 \sec{\left(x \right)}}\right)$$
 1
----
2*pi
$$\frac{1}{2 \pi}$$
= 0.159154943091895
     /cot(pi*x)*sin(x)\
lim |----------------|
x->0-\    2*sec(x)    /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} \cot{\left(\pi x \right)}}{2 \sec{\left(x \right)}}\right)$$
 1
----
2*pi
$$\frac{1}{2 \pi}$$
= 0.159154943091895
= 0.159154943091895
Rapid solution [src]
 1
----
2*pi
$$\frac{1}{2 \pi}$$
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} \cot{\left(\pi x \right)}}{2 \sec{\left(x \right)}}\right) = \frac{1}{2 \pi}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cot{\left(\pi x \right)}}{2 \sec{\left(x \right)}}\right) = \frac{1}{2 \pi}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cot{\left(\pi x \right)}}{2 \sec{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)} \cot{\left(\pi x \right)}}{2 \sec{\left(x \right)}}\right) = -\infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)} \cot{\left(\pi x \right)}}{2 \sec{\left(x \right)}}\right) = \infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cot{\left(\pi x \right)}}{2 \sec{\left(x \right)}}\right)$$
More at x→-oo
0.159154943091895
0.159154943091895
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type

The above examples also contain:

• the modulus or absolute value: absolute(x) or |x|
• square roots sqrt(x),
cubic roots cbrt(x)
• trigonometric functions:
sinus sin(x), cosine cos(x), tangent tan(x), cotangent ctan(x)
• exponential functions and exponents exp(x)
• inverse trigonometric functions:
arcsine asin(x), arccosine acos(x), arctangent atan(x), arccotangent acot(x)
• natural logarithms ln(x),
decimal logarithms log(x)
• hyperbolic functions:
hyperbolic sine sh(x), hyperbolic cosine ch(x), hyperbolic tangent and cotangent tanh(x), ctanh(x)
• inverse hyperbolic functions:
hyperbolic arcsine asinh(x), hyperbolic arccosinus acosh(x), hyperbolic arctangent atanh(x), hyperbolic arccotangent acoth(x)
• other trigonometry and hyperbolic functions:
secant sec(x), cosecant csc(x), arcsecant asec(x), arccosecant acsc(x), hyperbolic secant sech(x), hyperbolic cosecant csch(x), hyperbolic arcsecant asech(x), hyperbolic arccosecant acsch(x)
• rounding functions:
round down floor(x), round up ceiling(x)
• the sign of a number:
sign(x)
• for probability theory:
the error function erf(x) (integral of probability), Laplace function laplace(x)
• Factorial of x:
x! or factorial(x)
• Gamma function gamma(x)
• Lambert's function LambertW(x)

The insertion rules

The following operations can be performed

2*x
- multiplication
3/x
- division
x^2
- squaring
x^3
- cubing
x^5
- raising to the power
x + 7
x - 6
- subtraction
Real numbers
insert as 7.5, no 7,5

Constants

pi
- number Pi
e
- the base of natural logarithm
i
- complex number
oo
- symbol of infinity
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