## Evaluating Limits

### Methods of evaluation of limits

We shall divide the problems of evaluation of limits in five categories.

### (1) Algebraic limits:

Let f(x) be an algebraic function and ‘a’ be a real number. Then is known as an algebraic limit.

**Direct substitution method:**If by direct substitution of the point in the given expression we get a finite number, then the number obtained is the limit of the given expression.**Factorisation method:**In this method, numerator and denominator are factorised. The common factors are cancelled and the rest outputs the results.**Rationalisation method:**Rationalisation is followed when we have fractional powers (like \(\frac { 1 }{ 2 } ,\frac { 1 }{ 3 }\) etc.) on expressions in numerator or denominator or in both. After rationalisation the terms are factorised which on cancellation gives the result.**Based on the form when x → ∞:**In this case expression should be expressed as a function 1/x and then after removing indeterminate form, (if it is there) replace 1/x by 0.

### (2) Trigonometric limits:

To evaluate trigonometric limit the following results are very important.

### (3) Logarithmic limits:

To evaluate the logarithmic limits we use following formulae:

### (4) Exponential limits:

**(i) Based on series expansion:**

To evaluate the exponential limits we use the following results:

**(ii) Based on the form 1 ^{∞}:** To evaluate the exponential form

**1**we use the following results.

^{∞}### (5) L-Hospital’s rule:

If f(x) and g(x) be two functions of x such that

Sometimes it may be necessary to repeat this process a number of times till our goal of evaluating limit is achieved.

### Evaluating Limits Problems with Solutions

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