# Category:Examples of Limits of Real Functions

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This category contains examples of Limit of Real Function.

Let $\openint a b$ be an open real interval.

Let $c \in \openint a b$.

Let $f: \openint a b \setminus \set c \to \R$ be a real function.

Let $L \in \R$.

### Definition 1

$\map f x$ **tends to the limit $L$ as $x$ tends to $c$** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$

where $\R_{>0}$ denotes the set of strictly positive real numbers.

### Definition 2

$\map f x$ **tends to the limit $L$ as $x$ tends to $c$** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: x \in \map {N_\delta} c \setminus \set c \implies \map f x \in \map {N_\epsilon} L$

where:

- $\map {N_\epsilon} L$ denotes the $\epsilon$-neighborhood of $L$
- $\map {N_\delta} c \setminus \set c$ denotes the deleted $\delta$-neighborhood of $c$
- $\R_{>0}$ denotes the set of strictly positive real numbers.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

## Pages in category "Examples of Limits of Real Functions"

The following 20 pages are in this category, out of 20 total.

### L

- Limit at Infinity of x^n
- Limit of (Cosine (X) - 1) over X at Zero
- Limit of Absolute Value
- Limit of nth Root of Sum of nth Powers equals Maximum
- Limit of Real Function/Examples
- Limit of Real Function/Examples/e^-1 over size x at 0
- Limit of Real Function/Examples/Identity Function with Discontinuity at 0
- Limit of Real Function/Examples/Root of x at 1
- Limit of Real Function/Examples/Sine of Reciprocal of x at 0
- Limit of Real Function/Examples/x times Sine of Reciprocal of x at 0
- Limit of Real Function/Examples/x times Sine of Reciprocal of x at 0/Proof
- Limit of Sine of X over X at Zero
- Limit of Tan X over X at Zero
- Limit of x to the x
- Limit to Infinity of Error Function

### O

- One-Sided Limit of Real Function/Examples
- One-Sided Limit of Real Function/Examples/e^-1 over x at 0 from Left
- One-Sided Limit of Real Function/Examples/e^-1 over x at 0 from Right
- One-Sided Limit of Real Function/Examples/Floor of 1 minus Size x
- One-Sided Limit of Real Function/Examples/Reciprocal of 1 + e to the Reciprocal of x