# What is the limit? #lim_(xrarr2) (cos(pi/x))/(x-2)#

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Without L'Hospital's rule see below.

#### Explanation:

# = sin((pi(x-2))/(2x))/(x-2)#

# = pi/(2x) (sin((pi(x-2))/(2x)))/((pi(x-2))/(2x)#

As

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# lim_(xrarr2) (cos(pi/x))/(x-2) = (pi) /4 #

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We seek:

# L = lim_(xrarr2) (cos(pi/x))/(x-2) #

**Method 1:**

Both the numerator or denominator are zero at

# lim_(x rarr a) f(x)/g(x) = lim_(x rarr a) (f'(x))/(g'(x)) #

And so:

# L = lim_(xrarr2) (d/dx cos(pi/x))/(d/dx x-2) #

# \ \ \ = lim_(xrarr2) (-sin(pi/x) * ( -pi/x^2) )/(1) #

# \ \ \ = lim_(xrarr2) (pi sin(pi/x) /x^2) #

# \ \ \ = pi sin(pi/2) /2^2 #

# \ \ \ = (pi) /4 #

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#### Explanation:

Since we get

L'Hospital's Rule:

I am assuming you know the trigonometric relationships, the power rule, and the chain rule.

We now plug 2 in the place of

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