According to limit rules, which operation can be performed without special restrictions?

According to limit rules in calculus, limits can be evaluated through a variety of operations, but some combinations may have restrictions, particularly when it comes to division by zero or indeterminate forms. However, the operation that can be performed without special restrictions is the addition and subtraction of limits.

When you take the limit of the sum or difference of two functions, you can simply take the limit of each function individually and then perform the addition or subtraction. This is due to the linearity property of limits, which states that:

[

\lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)

]

and

[

\lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)

]

This property confirms that addition and subtraction can freely be applied without encountering issues like indeterminate forms or undefined behaviors that might occur with multiplication or division when one of the limits involved approaches zero.

In contrast, multiplication and division operations can introduce complications, such as the potential for producing undefined expressions.