```
In [1]:
```import numpy as np
import sympy as sp
import pandas as pd
import math
import midterm as p1
import matplotlib.pyplot as plt
# Needed only in Jupyter to render properly in-notebook
%matplotlib inline

```
In [2]:
```p1.sequence(0.5, 2.5, 100)

```
```

The function is not graphable at r = 2.5

```
In [13]:
```p1.sequence(0.5, 3.2, 10)

```
Out[13]:
```

```
In [4]:
```p1.graph(0.5, 3.2, 100)

```
```

```
In [12]:
```p1.sequence(0.5, 3.5, 10)

```
Out[12]:
```

```
In [6]:
```p1.graph(0.5, 3.5, 100)

```
```

```
In [11]:
```p1.sequence(0.25, 3.2, 10)

```
Out[11]:
```

```
In [8]:
```p1.graph(0.25, 3.2, 100)

```
```

```
In [10]:
```p1.sequence(0.75, 3.2, 10)

```
Out[10]:
```

```
In [10]:
```p1.graph(0.75, 3.2, 100)

```
```

```
In [9]:
```p1.sequence(0.25, 3.5, 10)

```
Out[9]:
```

```
In [12]:
```p1.graph(0.25, 3.5, 100)

```
```

```
In [8]:
```p1.sequence(0.75, 3.5, 10)

```
Out[8]:
```

```
In [14]:
```p1.graph(0.75, 3.5, 100)

```
```

```
In [7]:
```p1.sequence(0.5, 3.5441, 1000000)[-9:-1]

```
Out[7]:
```

```
In [51]:
```p1.graph(0.5, 3.5441, 100)

```
```

```
In [6]:
```p1.sequence(0.75, 3.5441, 1000000)[-9:-1]

```
Out[6]:
```

```
In [5]:
```p1.sequence(0.5, 3.5699, 1000000)[-33:-1]

```
Out[5]:
```

```
In [18]:
```p1.graph(0.5, 3.5699, 100)

```
```

```
In [4]:
```p1.sequence(0.75, 3.5699, 1000000)[-9:-1]

```
Out[4]:
```

```
In [10]:
```p1.sequence(0.5, 3.57, 1000000)[-33:-1]

```
Out[10]:
```

```
In [28]:
```p1.graph(0.5, 3.57, 100)

```
```

```
In [2]:
```p1.sequence(0.2, 3.57, 100000)[-8:-1]

```
Out[2]:
```

```
In [5]:
```p1.asymptote_graph(0.5, 2.9, 4, 0.0001, 200)

```
```

The following are graphs zoomed into specific regions of the graph above:

```
In [7]:
```p1.zoom_graph(0.5, 2.9, 4, 0.0001, 1000, [3.54, 3.55, 0.78, 0.9])

```
```

```
In [11]:
```p1.zoom_graph(0.5, 2.9, 4, 0.0001, 1000, [3.568, 3.572, 0.33, 0.39])

```
```

```
In [ ]:
```