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Find
lim_(h rarr0)(5arcsin((2)/(3)+h)-5arcsin((2)/(3)))/(h).
Choose 1 answer:
(A)
3sqrt5
(B)
(sqrt5)/(3)
(C)
(3sqrt5)/(5)
(D) The limit doesn't exist

Find $\lim _{h \rightarrow 0} \frac{5 \arcsin \left(\frac{2}{3}+h\right)-5 \arcsin \left(\frac{2}{3}\right)}{h}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $3 \sqrt{5}$ $\newline$ (B) $\frac{\sqrt{5}}{3}$ $\newline$ (C) $\frac{3 \sqrt{5}}{5}$ $\newline$ (D) The limit doesn't exist

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Find limits of polynomials and rational functions

Full solution

Q. Find

\lim _{h \rightarrow 0} \frac{5 \arcsin \left(\frac{2}{3}+h\right)-5 \arcsin \left(\frac{2}{3}\right)}{h}

.

\newline

Choose

1

answer:

\newline

(A)

3 \sqrt{5}

\newline

(B)

\frac{\sqrt{5}}{3}

\newline

(C)

\frac{3 \sqrt{5}}{5}

\newline

(D) The limit doesn't exist

Recognize Limit Problem: First, let's recognize that this is a limit problem involving the derivative of the $\arcsin$ function at a specific point.
Derivative of arcsin(x): The derivative of $\arcsin(x)$ is $\frac{1}{\sqrt{1-x^2}}$ . So, we can rewrite the limit as the derivative of $5\arcsin(x)$ at $x = \frac{2}{3}$ .
Calculate Derivative at $x = \frac{2}{3}$ : The derivative of $5\arcsin(x)$ is $5 \times \left(\frac{1}{\sqrt{1-x^2}}\right)$ . Plugging in $x = \frac{2}{3}$ , we get $5 \times \left(\frac{1}{\sqrt{1-\left(\frac{2}{3}\right)^2}}\right)$ .
Simplify Expression Inside Square Root: Simplify the expression inside the square root: $1 - \left(\frac{2}{3}\right)^2 = 1 - \frac{4}{9} = \frac{5}{9}$ .
Calculate Square Root: Now, calculate the square root: $\sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}$ .
Multiply by $5$ for Derivative: Multiply by $5$ to get the derivative: $5 \times \left(\frac{1}{\sqrt{5}/3}\right) = 5 \times \left(\frac{3}{\sqrt{5}}\right) = \frac{15}{\sqrt{5}}$ .
Rationalize Denominator: Simplify the expression by rationalizing the denominator: $(\frac{15}{\sqrt{5}}) \cdot (\frac{\sqrt{5}}{\sqrt{5}}) = \frac{15\sqrt{5}}{5}$ .
Final Simplification: Finally, simplify the fraction: $(15\sqrt{5})/5 = 3\sqrt{5}$ .

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