### Analysis of a real function

We will find and study:

- the domain and zeroes
- continuity, limits in the points in which function is not continuous and at the ends of intervals,
- asymptotes,
- differentiability and derivatives,
- intervals of monotonicity and extrema
- second derivative, convexity, inflection points
- the table of the function,
- parity, periodicity,
- sketch of the graph,
- range.

### An example

We will study .

#### the domain and zeroes

The denominator , if . Therefore, . Moreover, iff .

#### Continuity, limits in the points in which function is not continuous and at the ends of intervals,

The function is continuous on and .

Furthermore:

#### Asymptotes

Therefore we have vertical asymptotes i . There are no horizontal asymptotes.

We check oblique asymptotes:

Therefore, is the right oblique asymptote.

Therefore, is also the left oblique asymptote.

#### Differentiability and derivatives

The funtion in differentiable on the whole domain and:

#### Intervals of monotonicity and extrema

if or . Therefore:

- on we have , so decreases,
- on we have , so increases,
- on we have , so increases,
- on we have , so increases,
- on we have , so increases,
- on we have , so decreases.

Therefore, in the function has its local minimum, and in its local maximum.

#### Second derivative, convexity, inflection points

- on we have , so is convex,
- on we have , so is concave,
- on we have , so is convex,
- on we have , so is concave.

Therefore is an inflection point.

#### The table of the function

decreasing, convex | |||

local minimum | |||

increasing, convex | |||

– | – | vertical asymptote | |

increasing, concave | |||

inflection point | |||

increasing, convex | |||

– | – | vertical asymptote | |

increasing, concave | |||

local maximum | |||

decreasing, wklęsła. |

#### parity, periodicity

The function is odd, since . Therefore it is not even, because it is not a constant zero function. Obviously, it is not a periodic function.

#### Sketch of the graph

#### Range

Obviously, .