We intend to give a numerical and graphical approaches to the concept of limits using examples.
## Numerical Approach to Limits

**Example 1:** Let f(x) = 2 x + 2 and compute f(x) as x takes values closer to 1. We first consider values of x approaching 1 from the left (x < 1).

x |
f(x) |

0.5 |
3 |

0.8 |
3.6 |

0.9 |
3.8 |

0.95 |
3.9 |

0.99 |
3.98 |

0.999 |
3.998 |

0.9999 |
3.9998 |

0.99999 |
3.99998 |

We now consider x approaching 1 from the right (x > 1).

x |
f(x) |

1.5 |
5 |

1.2 |
4.4 |

1.1 |
4.2 |

1.05 |
4.1 |

1.01 |
4.02 |

1.001 |
4.002 |

1.0001 |
4.0002 |

1.00001 |
4.00002 |

In both cases as x approaches 1, f(x) approaches 4. Intuitively, we say that lim_{x→1}
f(x) = 4.

NOTE: We are talking about the values that f(x) takes when x gets closer to 1 and not f(1). In fact we may talk about the limit of f(x) as x approaches a even when f(a) is undefined.

**Example 2:** Let g(x) = sin x / x and compute g(x) as x takes values closer to 0. We consider values of x approaching 0 from the left (x < 0) and values of x approaching 0 from the right (x > 0).

x |
g(x) |

-0.5 |
0.9588 |

-0.2 |
0.993346 |

-0.1 |
0.998334 |

-0.01 |
0.999983 |

-0.001 |
0.999999 |

x |
g(x) |

0.5 |
0.9588 |

0.2 |
0.993346 |

0.1 |
0.998334 |

0.01 |
0.999983 |

0.001 |
0.999999 |

Here we say that lim_{x→0}
g(x) = 1. Note that g(0) is undefined.

## Graphical Approach to Limits

**Example 3:**

The graph below shows that as x approaches 1 from the left, y = f(x) approaches 2 and this can be written as

lim_{x→1-} f(x) = 2

As x approaches 1 from the right, y = f(x) approaches 4 and this can be written as

lim_{x→1+} f(x) = 4

Note that the left and right hand limits and f(1) = 3 are all different.

**Example 4:**

This graph shows that

lim_{x→1-} f(x) = 2

As x approaches 1 from the right, y = f(x) approaches 4 and this can be written as

lim_{x→1+} f(x) = 4

Note that the left hand limit and f(1) = 2 are equal.

**Example 5:**

This graph shows that

lim_{x→0-} f(x) = 1

and

lim_{x→0+} f(x) = 1

Note that the left and right hand limits are equal and we cvan write

lim_{x→0} f(x) = 1

In this example, the limit when x approaches 0 is equal to f(0) = 1.

**Example 6:**

This graph shows that as x approaches - 2 from the left, f(x) gets smaller and smaller without bound and there is no limit. We write

lim_{x→-2-} f(x) = - ∞

As x approaches - 2 from the right, f(x) gets larger and larger without bound and there is no limit. We write

lim_{x→-2+} f(x) = + ∞

Note that - ∞ and + ∞ are symbols and not numbers. These are symbols used to indicate that the limit does not exist.

**Example 7:**

The graph below shows a periodic function whose range is given by the interval [-1 , 1]. If x is allowed to increase without bound, f(x) take values within [-1 , 1] and has no limit. This can be written

lim_{x→ + ∞} f(x) = does not exist

If x is allowed to decrease without bound, f(x) take values within [-1 , 1] and has no limit again. This can be written

lim_{x→ - ∞ f(x) = does not exist
}

**Example 8:**

If x is allowed to increase without bound, f(x) in the graph below approaches 2. This can be written

lim_{x→ + ∞} f(x) = 2

If x is allowed to decrease without bound, f(x) approaches 2. This can be written

lim_{x→ - ∞} f(x) = 2

More on limits

Calculus Tutorials and Problems