Get the answer to "Evaluate limy→0(x+y)sec⁡(x+y)−xsec⁡xy"

Sage Fry

Answered question

2022-04-05

Evaluate

lim_{y \to 0} \frac{(x + y) \sec (x + y) - x \sec x}{y}

Answer & Explanation

Wilson Rivas

Beginner2022-04-06Added 12 answers

lim_{y \to 0} \frac{(x + y) \sec (x + y) - x \sec x}{y} = lim_{y \to 0} \frac{x \cos x (1 - \cos y) + y \cos y + x \sin x \sin y}{y \cos y \cos (x + y)}

= lim_{y \to 0} \underset{\to 0}{\underset{⏟}{\frac{(1 - \cos y)}{y}}} \frac{x \cos x}{\cos y \cos (x + y)}

+ lim_{y \to 0} \frac{1}{\cos (x + y)}

+ lim_{y \to 0} \frac{\sin y}{y} \frac{x \sin x}{\cos y \cos (x + y)}

= \frac{1}{\cos (x)} + \frac{x \sin x}{\cos (x)}

Nathanial Carey

Beginner2022-04-07Added 12 answers

Note that as $y \to 0$
$\frac{(x + y) \sec (x + y) - x \sec x}{y} = \frac{(x + y) \cos (x) - x \cos (x + y)}{y \cos (x + y) \cos (x)}$
$= \frac{x \cos (x) (1 - \cos (y)) + y \cos (x) + x \sin (x) \sin (y)}{y \cos (x + y) \cos (x)}$
$= \frac{x \cos (x)}{\cos (x + y) \cos (x)} \cdot \frac{1 - \cos (y)}{y} + \frac{1}{\cos (x + y)}$
$+ \frac{x \sin (x)}{\cos (x + y) \cos (x)} \cdot \frac{\sin (y)}{y}$
$\to 0 + \frac{1}{\cos (x)} + \frac{x \sin x}{\cos^{2} (x)} = \sec (x) (1 + x \tan (x))$
where we used
$\frac{1 - \cos (y)}{y} \to 0, \cos (x + y) \to \cos (x), \frac{\sin (y)}{y} \to 1 .$

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