How to show sec−1(1+x)=O(x12) as x→0According to the definition, I have to find the positive constant C, i.e., I want to obtain:limx→0|sec−1(1+x)x12|=CNow use the L'hospital's rule:limx→0|sec−1(1+x)x12|=limx→0|1(1+x)(1+x)2−112x−12|=limx→0|2x12(1+x)(1+x)2−1|

limx→0|2x12(1+x)(1+x)2−1|=limx→0|2x12(1+x)x2+2x|=limx→0|2x12(1+x){xx+2}|==limx→02x+2==2

Resolved: "How to show sec−1(1+x)=O(x12) as x→0According to the..."

Kasey Castillo

2022-04-07

How to show

\sec^{- 1} (1 + x) = O (x^{\frac{1}{2}})

x \to 0

According to the definition, I have to find the positive constant C, i.e., I want to obtain:

lim_{x \to 0} | \frac{\sec^{- 1} (1 + x)}{x^{\frac{1}{2}}} | = C

Now use the L'hospital's rule:

lim_{x \to 0} | \frac{\sec^{- 1} (1 + x)}{x^{\frac{1}{2}}} | = lim_{x \to 0} | \frac{\frac{1}{(1 + x) \sqrt{{(1 + x)}^{2} - 1}}}{\frac{1}{2} x^{- \frac{1}{2}}} | = lim_{x \to 0} | \frac{2 x^{\frac{1}{2}}}{(1 + x) \sqrt{{(1 + x)}^{2} - 1}} |

mislifola5vo

Beginner2022-04-08Added 11 answers

lim_{x \to 0} | \frac{2 x^{\frac{1}{2}}}{(1 + x) \sqrt{{(1 + x)}^{2} - 1}} | = lim_{x \to 0} | \frac{2 x^{\frac{1}{2}}}{(1 + x) \sqrt{x^{2} + 2 x}} | = lim_{x \to 0} | \frac{2 x^{\frac{1}{2}}}{(1 + x) {\sqrt{x} \sqrt{x + 2}}} | =

= lim_{x \to 0} \frac{2}{\sqrt{x + 2}} =

= \sqrt{2}

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