What is the limit as x approaches pi-of x/(cos(x)+1) ?

The limit as x approaches pi-of x/(cos(x)+1) is infinity

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limit as x approaches pi-of x/(cos(x)+1)

x \to π - lim (\frac{x}{cos ( x ) + 1})

\infty

Solution steps

x \to π - lim (\frac{x}{cos ( x ) + 1})

Use the following identity:

cos (x) = 1 - 2 sin^{2} (\frac{x}{2})

= x \to π - lim (\frac{x}{1 - 2 sin ^{2} ( \frac{x}{2} ) + 1})

Simplify

\frac{x}{1 - 2 sin ^{2} ( \frac{x}{2} ) + 1} : \frac{1}{2} x sec^{2} (\frac{x}{2})

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

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= \frac{1}{2} \cdot x \to π - lim (x sec^{2} (\frac{x}{2}))

x \to a lim [f (x) \cdot g (x)] = x \to a lim f (x) \cdot x \to a lim g (x)

With the exception of indeterminate form

= \frac{1}{2} \cdot x \to π - lim (x) \cdot x \to π - lim (sec^{2} (\frac{x}{2}))

x \to π - lim (x) = π

x \to π - lim (sec^{2} (\frac{x}{2})) = \infty

= \frac{1}{2} π \cdot \infty

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

What is the limit as x approaches pi-of x/(cos(x)+1) ?
The limit as x approaches pi-of x/(cos(x)+1) is infinity