# The limit of a function as x tends to infinity

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The limit of a function as *x *tends to infinity

In this unit, we explain what it means when the Limit of a function tends to infinity, at least infinity, or to a true bond, while x tends to infinity or infinity. We also explain what it means that a function tends to be real, while x tends to a certain real number.

In any case, we provide an example of a function that does not reach a limit. To master the techniques described here, it is important that you do many exercises to make it second nature.

In this example, we use Python programming to check the limit when x goes to infinity. The limit is a value that a function "approaches" because the input approaches some value. Limits are integral to calculus and are defined by using continuity, derivatives, and integrals.

**1. Discuss the limit of the following function when**

*x*→∞`import numpy as np from sympy import * x = symbols('x') init_printing(use_unicode = True) print(limit (sin(1/x),x,oo,'+')) plot(sin(1/x))`

Output: 0

The limit of a function as x tends to infinity |

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**2. Discuss the limit of the following function when**

*x*→∞

```
import numpy as np
from sympy import *
x = symbols('x')
init_printing(use_unicode = True)
print(limit (exp(1)/x,x,oo,'+'))
plot(exp(1)/x)
```**Output: 0**

The limit of a function as x tends to infinity |

**3. Discuss the limit of the following function when**

*x*→∞

```
import numpy as np
from sympy import *
x = symbols('x')
init_printing(use_unicode = True)
print(limit (exp(-1)/x,x,oo))
plot(exp(-1)/x)
```**Output: 0**

The limit of a function as x tends to infinity |

4.Discuss the limit of the following function whenx→∞`import numpy as np from sympy import * x = symbols('x') init_printing(use_unicode = True) print(limit (exp(x)/x,x,oo)) plot(exp(x)/x)`

Output: oo

The limit of a function as x tends to infinity |

5.Discuss the limit of the following function whenx→∞

```
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```
import numpy as np
from sympy import *
x = symbols('x')
init_printing(use_unicode = True)
print(limit (exp(-x)/x,x,oo))
plot(exp(-x)/x))
```

**The limit of a function as x tends to infinity**

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6. Discuss the limit of the following function whenx→∞`import numpy as np from sympy import * x = symbols('x') init_printing(use_unicode = True) print(limit (x**2*sin(1/x),x,oo,'+')) plot(x**2*sin(1/x))`

Output: oo

The limit of a function as x tends to infinity7. Discuss the limit of the following function whenx→∞import numpy as np from sympy import * x = symbols('x') init_printing(use_unicode = True) print(limit (x/(1+x),x,oo,'+')) plot(x/(1+x))Output:1

The limit of a function as x tends to infinity