# OP Malhotra Indeterminate Form Of Limits S.Chand ISC Class-12 Maths Solutions Ch-9

OP Malhotra **Indeterminate Form Of Limits** S.Chand ISC Class-12 Maths Solutions Ch-9. Step by step Solutions of OP Malhotra SK Gupta, Anubhuti Gangal S.Chand ISC Class-12 Mathematics with Exe-9(a), Exe-9(b), and Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

## OP Malhotra Indeterminate Form Of Limits S.Chand ISC Class-12 Maths Solutions Ch-9

Class: | 12th |

Subject: | Mathematics |

Chapter : | Ch-9 Indeterminate Form Of Limits of Section -A |

Board | ISC |

Writer | OP Malhotra, SK Gupta, Anubhuti Gangal |

Publications | S.Chand Publications 2020-21 |

**-: Included Topics :- **

Exe-9(a)

Exe-9(b)

Chapter Test

### OP Malhotra Indeterminate Form Of Limits S.Chand ISC Class-12 Maths Solutions Ch-9

**Indeterminate Form :**

In Mathematics, we cannot be able to find solutions for some form of Mathematical expressions. Such expressions are called indeterminate forms. In most of the cases, the indeterminate form occurs while taking the ratio of two functions, such that both of the function approaches to zero in the limit. Such cases are called “indeterminate form 0/0”. Similarly, the indeterminant form can be obtained in the addition, subtraction, multiplication, exponential operations also.

**Indeterminate Forms of Limits :**

Some forms of limits are called indeterminate if the limiting behaviour of individual parts of the given expression is not able to determine the overall limit.

**Indeterminate Forms List :**

Some of the indeterminate forms with conditions and transformation are given below:

Indeterminate form |
Conditions |

0/0 | lim_{x→cf}(x)=0,lim _{x→c }g(x)=0 |

∞/∞ | Lim _{x→c} f(x)=∞,lim _{x→c} g(x)=∞ |

0.∞ | Lim _{x→c} f(x)=0,lim _{x→cg} (x)=∞ |

∞-∞ | Lim _{x→c} f(x)=1,lim _{x→c} g(x)=∞ |

0^{0} |
Lim _{x→c} f(x)=0+,lim _{x→c} g(x)=0 |

1^{∞} |
Lim _{x→c} f(x)=∞,lim _{x→c} g(x)=∞ |

∞^{0} |
Lim _{x→c} f(x)=∞,lim _{x→c} g(x)=0 |

**Exe-9(a)**

### Indeterminate Form Of Limits S.Chand ISC Class-12 Maths Solutions Ch-9

**Find by applying L’ Hospital’s Rule the following limits :**

Question 1:

lim……………..

Question 2:

………………..

………………….

………………….

Question 28:

lim {1/x² – cot x/x}

……………….

**How to Evaluate Indeterminate Forms :-**

**There are three methods used to evaluate indeterminate forms. They are:**

**Factoring Method (0/0 form)**

In the factoring method, the expressions are factorized to their maximum simplest form. After that, the limit value should be substituted.

**L Hospital’s Rule (0/0 or ∞/∞ form) :**

In this method, the derivative of each term is taken in each step successively until at least one of the terms becomes free of the variable. It means that at least one term becomes constant.

**Division of Each Term by Highest Power of Variable (∞/∞ form) :**

In this method, each term in numerator and denominator is divided by the variable of the highest power in the expression, and then, the limit value is obtained.

**Exe-9(b)**

### OP Malhotra Indeterminate Form Of Limits S.Chand ISC Class-12 Maths Solutions Ch-9

**Find by applying L’ Hospital’s Rule the following limits :**

Question 1:

lim xe^{x }– log(1-x)/x²

Question 2:

…………………..

……………………

…………………….

Question 20:

lim [e^{x }– e^{-x }– 2x/x – sin x]

…………….

**Chapter Test**

Indeterminate Form Of Limits S.Chand ISC Class-12 Maths Solutions Ch-9

Question 1:

lim e^{2x }– 1/x

Question 2:

………………….

……………………

……………………

Question 10:

lim [tan(x² – 1)/x – 1]

-: End of Indeterminate Form Of Limits **OP Malhotra S. Chand **ISC Class-12 Maths Solution :-

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