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For what values of x, if any, does #f(x) = secpix # have vertical asymptotes?

Answer 1

Paisley Johnson

# x = ((2k+1) )/2, qquad k in mathbf(Z) #

for vertical asymptotes we look for a denominator of 0

here we have

#f = sec pi x = 1/ (cos pi x)#

the denom will be zero at #pi x = pi/2, (3pi)/2 , ...., ((2k+1) pi)/2, qquad k in mathbf(Z) #

ie # x = ((2k+1) )/2, qquad k in mathbf(Z) #

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Answer 2

Christopher Agresta

The function f(x) = secpix has vertical asymptotes when the secant function approaches positive or negative infinity. This occurs when the cosine of the angle pix is equal to zero. Therefore, the values of x for which f(x) has vertical asymptotes are given by x = (2n + 1)π/2, where n is an integer.

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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.