# Indeterminate Forms

In calculus and other divisions of math study, an indeterminate forms is arithmetical expressions find in the circumstance of limits. Limits relating arithmetical operations are often execute by restoring sub expressions by their limits; if the term find after this replacement does not give sufficient information to establish the creative limit, it is recognized as an indeterminate form. The indeterminate forms contain 00, 0/0, 1oo -oo , oo/oo , 0 × oo , and oo 0.

## List of Indeterminate forms

Below are the list of Indeterminate forms -

Indeterminate forms of limits 1:

0/0

Conditions

 \lim_{x \to c} f(x) = 0, \lim_{x \to c} g(x) = 0

Transformation to oo/oo

 \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{1/g(x)}{1/f(x)}

Indeterminate forms of limits 2:

oo/oo

Conditions

\lim_{x \to c} f(x) = \infty, \lim_{x \to c} g(x) = \infty

Transformation to 0/0

 \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{1/g(x)}{1/f(x)}

Indeterminate forms of limits 3:

 0 xx oo

Conditions

 \lim_{x \to c} f(x) = 0, \lim_{x \to c} g(x) = \infty

Transformation to 0/0

\lim_{x \to c} f(x)g(x) = \lim_{x \to c} \frac{f(x)}{1/g(x)}

Transformation to oo/oo

\lim_{x \to c} f(x)g(x) = \lim_{x \to c} \frac{g(x)}{1/f(x)}

Indeterminate forms of limits 4:

1^oo

Conditions

 \lim_{x \to c} f(x) = 1, \lim_{x \to c} g(x) = \infty

Transformation to 0/0

\lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{\ln f(x)}{1/g(x)}

Transformation to oo/oo

\lim_{x \to c} f(x)^{g(x)} = \lim_{x \to c} \frac{g(x)}{1/\ln f(x)}

Indeterminate forms of limits 5:

0O

Conditions

\lim_{x \to c} f(x) = 0^+, \lim_{x \to c} g(x) = 0

Transformation to 0/0

 \lim_{x \to c} f(x)^{g(x)} = \lim_{x \to c} \frac{g(x)}{1/\ln f(x)}

Transformation to oo/oo

 \lim_{x \to c} f(x)^{g(x)} = \lim_{x \to c} \frac{\ln f(x)}{1/g(x)}

Indeterminate forms of limits 6:

ooO

Conditions

\lim_{x \to c} f(x) = \infty, \lim_{x \to c} g(x) = 0

Transformation to 0/0

\lim_{x \to c} f(x)^{g(x)} = \lim_{x \to c} \frac{g(x)}{1/\ln f(x)}

Transformation to oo/oo

\lim_{x \to c} f(x)^{g(x)} = \lim_{x \to c} \frac{\ln f(x)}{1/g(x)}

Indeterminate forms of limits 7:

oo-oo

Conditions

\lim_{x \to c} f(x) = \infty, \lim_{x \to c} g(x) = \infty

Transformation to 0/0

\lim_{x \to c} (f(x) - g(x)) = \lim_{x \to c} \frac{1/g(x) - 1/f(x)}{1/(f(x)g(x))}

Transformation to oo/oo

 \lim_{x \to c} (f(x) - g(x)) = \lim_{x \to c} \frac{e^{f(x)}}{e^{g(x)}}

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## Indeterminate forms Examples

Below see the example on indeterminate forms -

## Example :

The indeterminate form 0/0 is mainly general in calculus since it happen in the estimate of derivatives by limit definition.

As state above,

\lim_{x \to 0} \frac{x}{x} = 1,

While

 \lim_{x \to 0} \frac{x^{2}}{x} = 0.

This is sufficient to prove that 0/0 is an indeterminate form.

## Indeterminate forms limits - Example 2:

 \lim_{x \to 0} \frac{\sin(x)}{x} = 1 and \lim_{x \to 49} \frac{x - 49}{\sqrt{x}\, - 7} = 14 .

Through replacement of the number those x moves toward into some of these terms guide to the indeterminate form 0/0, excluding the limits get a lot of different values. In really, any preferred value A can be get for this indeterminate form as follows:

 \lim_{x \to 0} \frac{Ax}{x} = A.

As well, the value perpetuity can also be getting in the sense of deviation to infinity:

 \lim_{x \to 0} \frac{x}{x^3} = \infty.