limx---->0 [arcsinx/x]+[2arcsinx/2x]+[3arcsinx/3x]+[4arcsinx/4x] is equal to
(a)30
(b)0
(c)26
(d)10

To solve the limit expression you provided, we need to analyze the behavior of each term as \( x \) approaches 0. The expression is:

lim_x→0 \[ \frac{\arcsin x}{x} + \frac{2 \arcsin x}{2x} + \frac{3 \arcsin x}{3x} + \frac{4 \arcsin x}{4x} \]

We can simplify each term in the limit. Notice that each term has the form:

• \[ \frac{n \arcsin x}{nx} = \frac{\arcsin x}{x} \] for \( n = 1, 2, 3, 4 \).

Thus, the entire expression can be rewritten as:

lim_x→0 \[ 4 \cdot \frac{\arcsin x}{x} \]

Now, we need to evaluate the limit of \(\frac{\arcsin x}{x}\) as \( x \) approaches 0. A well-known limit in calculus states:

lim_x→0 \[ \frac{\arcsin x}{x} = 1 \]

Using this result, we can substitute back into our expression:

lim_x→0 \[ 4 \cdot \frac{\arcsin x}{x} = 4 \cdot 1 = 4 \]

However, we must remember that we have four identical terms contributing to the limit. Therefore, we multiply the limit by 4:

4 \cdot 1 = 4

Now, let's summarize what we have:

• The limit of \(\frac{\arcsin x}{x}\) as \( x \) approaches 0 is 1.
• Since there are four identical terms, we multiply the limit by 4.

Thus, the final result of the limit expression is:

4

Since none of the provided options (30, 0, 26, 10) match our calculated limit of 4, it appears there may be a misunderstanding or miscommunication regarding the question or options. If you have any further details or clarifications, feel free to share!