Question

What are the properties of limits?

Answer 1

The properties of limits:

A limit is a value that the function approaches the output value for a given input value.

Let $c$ and $k$ be real numbers then,

1. Sum rule: The limit of the sum of two functions is the sum of the individual limits.

If $\underset{x\to c}{\lim }\left[f\left(x\right)+g\left(x\right)\right]=\underset{x\to c}{\lim }f\left(x\right)+\underset{x\to c}{\lim }g\left(x\right)$

2. Difference rule: The limit of the difference between two functions is the difference of the individual limits.

If $\underset{x\to c}{\lim }\left[f\left(x\right)-g\left(x\right)\right]=\underset{x\to c}{\lim }f\left(x\right)-\underset{x\to c}{\lim }g\left(x\right)$

3. Product rule: The limit of the product of two functions is the product of the individual limits.

If $\underset{x\to c}{\lim }\left[f\left(x\right)·g\left(x\right)\right]=\underset{x\to c}{\lim }f\left(x\right)·\underset{x\to c}{\lim }g\left(x\right)$

4. Constant Multiple rule: The limit of the product of a constant and a function is the constant times the limit of the function

If $\underset{x\to c}{\lim }\left[k·f\left(x\right)\right]=k·\underset{x\to c}{\lim }f\left(x\right)$

5. Quotient rule: The limit of the quotient of two functions is the quotient of the individual limits provided the denominator is not equal to zero.

$\underset{x\to c}{\lim }\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{\underset{x\to c}{\lim }f\left(x\right)}{\underset{x\to c}{\lim }g\left(x\right)},\underset{x\to c}{\lim }g\left(x\right)\ne 0$

6. Power rule: The limit of rational power of a given function is the power of limit of the function, provided they are real numbers

$\underset{x\to c}{\lim }\left[f{\left(x\right)}^{\frac{r}{s}}\right]={\left[\underset{x\to c}{\lim }f\left(x\right)\right]}^{\frac{r}{s}},r,s$are integers $,s\ne 0$

7. Law of root: The limit of $n^{th}$root of a function is the $n^{th}$root of the limit of the function

$\underset{x\to c}{\lim }\sqrt[n]{f\left(x\right)}=\sqrt[n]{\underset{x\to c}{\lim }f\left(x\right)},n$is an integer

8. Law of constant: the limit of a constant is the constant itself.

$\underset{x\to c}{\lim }u=u$