Limit of Real Function/Examples

Examples of Limits of Real Functions

Example: Identity Function with $1$ at $0$

Let $f$ be the real function defined as:

$\map f x = \begin {cases} x & : x \ne 0 \\ 1 & : x = 0 \end {cases}$

Then:

$\ds \lim_{x \mathop \to 0} \map f x = 0$

Example: $\sqrt x$ at $1$

$\ds \lim_{x \mathop \to 1} \sqrt x = 1$

Example: $e^{-1 / \size x}$ at $0$

$\ds \lim_{x \mathop \to 0} e^{-1 / \size x} = 0$

Example: $\map \sin {\dfrac 1 x}$ at $0$

Let:

$\map f x = \map \sin {\dfrac 1 x}$

Then:

$\ds \lim_{x \mathop \to 0} \map f x$

does not exist.

Example: $x \map \sin {\dfrac 1 x}$ at $0$

Let:

$\map f x = x \map \sin {\dfrac 1 x}$

Then:

$\ds \lim_{x \mathop \to 0} \map f x = 0$