JUMP TO TOPIC

# Indeterminate|Definition & Meaning

## Definition

The word **“indeterminate”** represents an **anonymous** value. The indeterminate state is a **Mathematical term** that points out that we are not able to decide the **actual** value even after **substituting** the limits.

An indeterminate is an **additional** expression for a variable. An indeterminate equation has **greater** than **one** solution. For instance x + y = 2, Without **additional** data we are unable to say exactly what x and y **actually** are. So we can say that An **indeterminate** expression cannot be exactly **determined. **

Both of the **functions** approach **zero** in the limit, **mainly** when the indeterminate state appears while assuming the **ratio** of **two** functional operations. Such **forms** are called **“indeterminate form 0/0”.** Again, the indeterminate form can be acquired in **addition,** **multiplication, subtraction,** and also **exponential** operations.

## Indeterminate Forms of Limits

If a **mathematical** form is not **definitively** or **exactly** determined, it can also be expressed to be indeterminate. Various forms of limits are **communicated** to be **indeterminate** when simply understanding the limiting conduct of particular aspects of the **expression** is **not adequate** to really decide the **all-around limit.**

For instance, a limit of the form 0/0, i.e., $lim_{x \rightarrow 0} f(x)/g(x)$ where $lim_{x \rightarrow 0}f(x)= lim_{x \rightarrow 0}g(x)=0$, is **indeterminate.** the **limiting** behavior of the **mixture** of the **two** functions (e.g., $lim_{x \rightarrow 0} x/x=1$, while $lim_{x \rightarrow 0}x^2/x=0)$ **controls** the value of the all-around limit.

There are **seven** indeterminate forms involving 0, 1, and infinity, stated as follows.

## Importance of Indeterminate Forms in Calculus

If we dig better into the **delicacies** of indeterminate forms, we can **realize** how they can be **used** to define the **attributes** of any **procedure** at any given limit in any sort of way. The **shortage** of proper **data** demarcates an indefinite shape. Regardless, there is **periodically** too much information that entails lessening an answer to a **distinctive** strategy. **Calculus** is messed up with indeterminate forms.

In **decree** to **decode** for a limit when functioning with ratios like **1/0** and **0/0** or with an epoch of any kind, we will nearly definitely require to use another **theorem,** such as **L’Hopital’s Rule.** The reason that indeterminate forms are often **resolvable** is they can provide some light on a value that was **formerly unidentified.**

Assume the **0/0** fraction. Anybody with a **primary** knowledge of **fractions** would easily guess that this is **equivalent** to **one** as the **numerator** corresponds to the **denominator,** which is equivalent to one. A better **applicable** approach would be to presume that the fraction is zero thoroughly**.** Similarly, a zero in the **denominator** could illustrate **infinity** or the lack of anything.

As a consequence, there are **multifarious** categories for this **ratio,** but there is no clear conqueror. Then why **are indeterminate forms** needed in **realistic-world** scenarios? The clear **answer** to this **problem** is that they are not **crucial** at the smallest and not in seclusion. Nevertheless, their **existence spreads** the spadework for us to improvise new **techniques** to decode indeterminate forms.

## Methods to Evaluate Indeterminate Forms

Following are the **techniques **that are used to solve the Indeterminate Forms.

### Factoring Method

This procedure is **commonly** operated for **0/0** form, and it demands **factoring** the provided equations to their **easiest** forms. The **limit** value is used to solve after the most uncomplicated term has been **emanated.**

### L’Hospital’s Rule

In the matter of an **indeterminate** form, the rule states that **before** applying the limit the best way to **decode** it is to distinguish the numerator and denominator **separately.** After each step, the **products** of the numerator and denominator are **analyzed** separately to inspect if they have become **unrestricted** of the variable, and as a result at least one of the terms remain **constant** or stable.

### Division by Highest Power

When the **indeterminate** form is usually introduced in the / state, this **methodology** is generally utilized. In this **procedure,** dividing both the **numerator** and **denominator** of the given term by the most **increased** power variable in the sum is the most **useful** route of action. After this whole procedure, the **limit value** is decided.

## Indeterminate Forms List

Examining all Indeterminate Forms **on an individual basis** below.

### Indeterminate Form 1

**Case:** 0/0

**Condition:**

\[lim_{x \rightarrow c} f(x) = 0, lim_{x \rightarrow c} g(x) = 0\]

**Transformation:**

Transformation to ∞/∞. Then it transforms into:

\[ lim_{x \rightarrow c} \dfrac{f(x )}{g(x )} = lim_{x \rightarrow c} \dfrac{1/g(x )}{1/f(x )}\]

### Indeterminate Form 2

**Case:** $\infty / \infty$

**Condition: **

\[lim_{x \rightarrow c} f(x) = \infty, lim_{x \rightarrow c} g(x) = \infty\]

**Transformation:**

Transformation to 0/0. After that, it becomes:

\[ lim_{x \rightarrow c} \dfrac{f(x )}{g(x )} = lim_{x \rightarrow c} \dfrac{1/g(x )}{1/f(x )}\]

### Indeterminate Form 3

**Case:** $ 0 \times \infty$

**Condition: **

\[lim_{x \rightarrow c} f(x) = 0, lim_{x \rightarrow c} g(x) = \infty\]

**Transformation:**

Transformation to 0/0, which results in the following:

\[ lim_{x \rightarrow c} f(x )g(x ) = lim_{x \rightarrow c} \dfrac{f(x )}{1/g(x )}\]

Transformation to $\dfrac{\infty}{\infty}$, resulting in:

\[ lim_{x \rightarrow c} f(x )g(x ) = lim_{x \rightarrow c} \dfrac{g(x )}{1/f(x )}\]

### Indeterminate Form 4

**Case:** $ 1^{\infty}$

**Condition: **

\[lim_{x \rightarrow c} f(x) = 1, lim_{x \rightarrow c} g(x) = \infty\]

**Transformation:**

Transformation to 0/0, after that, it becomes:

\[ lim_{x \rightarrow c} f(x )^{g(x )} = exp \space lim_{x \rightarrow c} \dfrac{lnf(x )}{1/g(x )}\]

Transformation to $\dfrac{\infty}{\infty}$, resulting in:

\[ lim_{x \rightarrow c} f(x )^{g(x )} = lim_{x \rightarrow c} \dfrac{g(x )}{1/lnf(x )}\]

### Indeterminate Form 5

**Case:** $ 0^0$

**Condition: **

\[lim_{x \rightarrow c} f(x) = 0^+, lim_{x \rightarrow c} g(x) = 0\]

**Transformation:**

Transformation to 0/0, then it converts to:

\[ lim_{x \rightarrow c} f(x )^{g(x )} = lim_{x \rightarrow c} \dfrac{g(x )}{1/lnf(x )}\]

Transformation to $\dfrac{\infty}{\infty}$. It transforms into:

\[ lim_{x \rightarrow c} f(x )^{g(x )} = lim_{x \rightarrow c} \dfrac{lnf(x )}{1/g(x )}\]

### Indeterminate Form 6

**Case:** $ \infty^0$

**Condition: **

\[lim_{x \rightarrow c} f(x) = \infty, lim_{x \rightarrow c} g(x) = 0\]

**Transformation:**

Transformation to 0/0. Then it develops into:

\[ lim_{x \rightarrow c} f(x )^{g(x )} = lim_{x \rightarrow c} \dfrac{g(x )}{1/lnf(x )}\]

Transformation to $\dfrac{\infty}{\infty}$. It turns into:

\[ lim_{x \rightarrow c} f(x )^{g(x )} = lim_{x \rightarrow c} \dfrac{lnf(x )}{1/g(x )}\]

### Indeterminate Form 7

**Case:** $ \infty – \infty$

**Condition: **

\[lim_{x \rightarrow c} f(x) = \infty, lim_{x \rightarrow c} g(x) = \infty\]

**Transformation:**

Transformation to 0/0. Then it turns to:

\[ lim_{x \rightarrow c} (f(x )-g(x )) = lim_{x \rightarrow c} \dfrac{|1/g(x)| – |1/f(x)|}{1/|f(x) g(x)|}\]

Transformation to $\dfrac{\infty}{\infty}$, resulting in:

\[ lim_{x \rightarrow c} (f(x )-g(x )) = lim_{x \rightarrow c} \dfrac{e^{f(x)}}{e^{g(x )}}\]

## Example of Solving an Indeterminate System

Consider the given expression:

### Solution

Let $f(x) = \sin(2x)$ and $g(x) = e^x +x$. Then:

\[ f'(x) = 2\cos(2x), \space g'(x)= e^x +1\]

\[ \lim{x \rightarrow 0} = \dfrac{f'(x)}{g(x)} = \lim{x \rightarrow 0} \dfrac{2\cos(2x)}{e^x +1}\]

Replacing limits:

\[ = \dfrac{2\cos(0)}{e^0 +1}\]

= 2/2 = 1

*All images/mathematical drawings were created with GeoGebra.*