How do you find the limit of #(x-pi/2)tanx# as #x->pi/2#?

Answer 1

Rewrite so we can use #lim_(thetararr0)theta/sintheta = 1#

#cos(pi/2-x) = sinx# (by co-function identity or difference formula for cosine)
Also #cos(-theta) = cos theta# (cosine is an even function)
So #cos(x-pi/2) = cos(pi/2-x) = sinx# and
#sin(pi/2-x) = cosx# (by co-function identity or difference formula for sine)
Also #sin(-theta) = -sin theta# (sine is an odd function)
So #sin(x-pi/2) = -sin(pi/2-x) = -cosx#and

Now we can rewrite

#lim_(xrarrpi/2)(pi/2-x)tanx = lim_(xrarrpi/2)(pi/2-x)sinx/cosx#
# = lim_(xrarrpi/2)(pi/2-x)cos(pi/2-x)/(-sin(pi/2-x))#
# = lim_(xrarrpi/2)-(pi/2-x)/(sin(pi/2-x)) * cos(pi/2-x)#
Notice that #lim_(xrarrpi/2)(pi/2-x)/(sin(pi/2-x))# is a version of #lim_(thetararr0)theta/sintheta #, so we get
# = -(1)(1) = -1#
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Answer 2

To find the limit of (x - π/2)tan(x) as x approaches π/2, we can use the concept of L'Hôpital's Rule.

First, we differentiate both the numerator and denominator with respect to x. The derivative of (x - π/2) is 1, and the derivative of tan(x) is sec^2(x).

Next, we substitute π/2 into the derivatives to get 1/(sec^2(π/2)). Since sec^2(π/2) is equal to 1, the limit simplifies to 1/1, which is equal to 1.

Therefore, the limit of (x - π/2)tan(x) as x approaches π/2 is 1.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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