Learn definition of infinite limit calculus with examples, one sided limits, and vertical asymptotes. Practice to find limits from a graph.

- Consider the function

\(f(x)=x^2+x+1\)

The values of \('f'\) for values of x which are near \(x = 1\), are as shown:

\(x\) | \(f(x)\) | \(x\) | \(f(x)\) |
---|---|---|---|

.7 | 2.19 | 1.3 | 3.99 |

.8 | 2.44 | 1.2 | 3.64 |

.9 | 2.71 | 1.1 | 3.31 |

.95 | 2.8525 | 1.05 | 3.1525 |

.99 | 2.9701 | 1.01 | 3.031 |

.999 | 2.997 | 1.001 | 3.003 |

Observe that as \(x\) approaches close to 1, the values of \(f(x)\) approach 3. We write this as

\(\lim\limits_{x\rightarrow 1}(x^2+x+1)=3\) (lim is short for the Limit)

- \(\lim\limits_{x\rightarrow a}f(x)=k\) will mean that as \(x\) takes values closer and closer to 'a', \(f(x)\) will take values closer and closer to k.
- We are not interested in what is happening at \(x=a\) in the above problem. In fact, function may not be defined at \(x=a\).

Note that \(x\rightarrow2\) means \(x\) takes values very close to 2, we now distinguish between the two cases when \(x\) is close to '2' but higher than '2' and lower than '2'.

So,

\(x\rightarrow a^+\) \(\Rightarrow\) \(x\) takes values close to 'a' but higher than 'a'

\(x\rightarrow a^-\) \(\Rightarrow\) \(x\) takes values close to 'a' but lower than 'a'

So,

\(x\rightarrow 3^-\) will mean \(x\) = 2.9999 and

\(x\rightarrow 3^+\) will mean \(x\) = 3.0001

- \(\lim\limits_{x\rightarrow a^-} f(x)=L\) will mean that values of f(x) approach L. When we take values of \(x\) closer to 'a' and lower than 'a' or simply the left hand limit of f(x) at \( x = a\) is L.
- Similarly, \(\lim\limits_{x\rightarrow a^+} f(x)=M\) will mean that right hand limit of f(x) at \( x = a\) is M.

Consider the graph of a function f(x):

\(\lim\limits_{x\rightarrow2^+}f(x)=3\) (because the height of graph to the immediate right of 2 is 3)

\(\lim\limits_{x\rightarrow2^-}f(x)=2\) (because the height of graph to the immediate left of 2 is 2).

\(f(2)=3\) (empty circle indicates that (2, 2) is not a part of the graph)

We say that

\(\lim\limits_{x\rightarrow a}f(x)\) exists if

\(\lim\limits_{x\rightarrow a^{-}}f(x)=\lim\limits_{x\rightarrow a^{+}}f(x)\)

or left hand limit at ''x = a" = right hand limit at ''x = a''.

To find the existence of the limit \(\lim\limits_{x\rightarrow a}f(x)\) we find

- R.H.L. = \(\lim\limits_{x\rightarrow a^+}f(x)\) (look at the height of graph to immediate right of a )
- L.H.L. = \(\lim\limits_{x\rightarrow a^-}f(x)\) (look at the height of graph to immediate left of a).

Compare both values, if they are not equal we say that limit doesn't exist. If they are equal we say limit exists and equals R.H.L. or L.H.L.

A \(\lim\limits_{x\rightarrow 1}f(x)\)

B \(\lim\limits_{x\rightarrow {-3}}f(x)\)

C \(\lim\limits_{x\rightarrow {3}}f(x)\)

D \(\lim\limits_{x\rightarrow {4}}f(x)\)

We sometimes say,

\(\lim\limits_{x\rightarrow a}\;f(x)=\infty\), this means that values of f(x) can be made as large as possible when values of x approach 'a', or values of f(x) increase without bounds when \(x\rightarrow a\).

- Note that \(\infty\) is not a number, its just a symbol, used to express a concept.
- Similarly, \(\lim\limits_{x\rightarrow a}\;f(x)=-\infty\) will mean that values of f(x) can be made as large negative as possible when values of x approach 'a'.

In the graph shown:

A \(\infty\)

B \(-\infty\)

C 2

D –3

The vertical line x = a is called the vertical asymptote of a function y = f(x) or a curve y = f(x) if at least one of following six is true.

(1) \(\lim\limits_{x\rightarrow a^+}\;f(x)=\infty\)

(2) \(\lim\limits_{x\rightarrow a^-}\;f(x)=\infty\)

(3) \(\lim\limits_{x\rightarrow a^+}\;f(x)=-\infty\)

(4) \(\lim\limits_{x\rightarrow a^-}\;f(x)=-\infty\)

(5) \(\lim\limits_{x\rightarrow a}\;f(x)=\infty\)

(6) \(\lim\limits_{x\rightarrow a}\;f(x)=-\infty\)

- To find the vertical asymptote of a curve y = f(x), put the denominator of f(x) = 0 and solve for x. If the solution is x = a then it is the vertical asymptote.

A \(x = 3, \,x = 1\)

B \(x = 4, \,x = 7\)

C \(x = –10, \,x = 8\)

D \(x = –5,\, x = 3\)