Solving Limit from the Left

Introduction for solving limit from the left:

While defining the limit of a function as x tends to c, we consider values off(x) when x is very close to c. The values of x may be greater or less than c. Ifwe restrict x to values less than c, then we say that x tends to c from below or from the left and write it symbolically as x ? c - 0 or simply x ? c-. The limit of f with this limit on x, is called the left hand limit. This is written as

Lf(c) =`lim_(x->c-)` f(x), provided the limit exists.

In this article we shall the Solving Limits is the way of education to the students through internet in which the tutor provides step by step solution to example problems.

Example Problems for Solving Online Limit from the Left:

Example 1 for solving online limit from the left:

solving `lim_(c->1)`` (c^2 - 1)/(c -1)` if it exists.

Solution:

Let us evaluate the left hand

When c ? 1-, put c = 1 - h, h > 0.

Then  - `lim_(c->1-)``(c^2 -1)/(c - 1)`

= `lim_(h->0)``((1 - h)^2 - 1)/(1 - h - 1)`

= `lim_(h->0)``(1 - 2h + h^2 - 1)/- h`

=  `lim_(h->0)` (2 - h) = limh ? 0 (2) -limh ? 0 (h) = 2 - 0 = 2

Example 2 for solving online limit from the left:

Solving  the left hand limits of the function at c= 4

f(c) =` | c- 4 |/(c - 4)` for c ? 4

0, for c = 4

Solution:

Again when c < 4, | c - 4 | = - (c - 4)

Therefore  `lim_(c->-4)` f(c) = `lim_(c->-4)``-(c - 4)/(c - 4)`

= `lim_(c->4-)`(- 1) = - 1

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Example 3 for Solving Online Limit from the Left:
solving this limit  `lim_(c->0)``( 3c + | c |)/(7c - 5 |c |)` , if it ecists.

Solution:

L f(0) =`lim_(c->0-)``(3c + | c |)/(7c - 5 |c |)`

=`lim_(c->0-)``(3c - c)/(7c - 5(- c))`

(since c < 0, | c | = - c)

=`lim_(c->0-)``(2c)/(12c)` =`lim_(c->0-)` `(1/6)` =`1/6.`