What is the limit as x approaches 0 of xtan(pi/2-x) ?

The limit as x approaches 0 of xtan(pi/2-x) is 1

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limit as x approaches 0 of xtan(pi/2-x)

x \to 0 lim (x tan (\frac{π}{2} - x))

1

Solution steps

x \to 0 lim (x tan (\frac{π}{2} - x))

Rewrite for L'Hopital

= x \to 0 lim \frac{x}{\frac{1}{t a n ( \frac{π}{2} - x )}}

Apply L'Hopital's Rule

: x \to 0 lim (\frac{1}{sec ^{2} ( x )})

= x \to 0 lim (\frac{1}{sec ^{2} ( x )})

Plug in the value

x = 0

= \frac{1}{sec ^{2} ( 0 )}

Simplify

\frac{1}{sec ^{2} ( 0 )} : 1

= 1

x \to 2 - lim (\frac{- 3}{x - 2})

x \to 0 lim (\frac{4 x - x ^{2}}{x})

x \to 0 lim (\frac{x}{2 x ^{2}})

x \to 0 + lim (\frac{e ^{1 + x} - e}{x})

x \to - 2 lim (x^{2} + 4)

What is the limit as x approaches 0 of xtan(pi/2-x) ?
The limit as x approaches 0 of xtan(pi/2-x) is 1