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 4 subcategories, out of 4 total.
L
- Limit of x to the x (3 P)
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