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Evaluating Limits

May 1, 2023 by Prasanna

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 Evaluating Limits 1 is known as an algebraic limit.

  1. 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.
  2. Factorisation method: In this method, numerator and denominator are factorised. The common factors are cancelled and the rest outputs the results.
  3. 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.
  4. 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.
Evaluating Limits 2

(3) Logarithmic limits:

To evaluate the logarithmic limits we use following formulae:
Evaluating Limits 3

(4) Exponential limits:

(i) Based on series expansion:
Evaluating Limits 4
To evaluate the exponential limits we use the following results:
Evaluating Limits 5
(ii) Based on the form 1∞: To evaluate the exponential form 1∞ we use the following results.
Evaluating Limits 6

(5) L-Hospital’s rule:

If f(x) and g(x) be two functions of x such that
Evaluating Limits 7
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

1.
Evaluating Limits 8
Solution:
Evaluating Limits 9
2.
Evaluating Limits 10
Solution:
Evaluating Limits 11
3.
Evaluating Limits 12
Solution:
Evaluating Limits 13
4.
Evaluating Limits 14
Solution:
Evaluating Limits 15
5.
Evaluating Limits 16
Solution:
Evaluating Limits 17
6.
Evaluating Limits 18
Solution:
Evaluating Limits 19
7.
Evaluating Limits 20
Solution:
Evaluating Limits 21
8.
Evaluating Limits 22
Solution:
Evaluating Limits 23

Filed Under: Mathematics Tagged With: Algebraic limits, Based on the form when x → ∞, Direct substitution method, Evaluating Limits, Evaluating Limits Problems with Solutions, Exponential limits, Factorisation method, L-Hospital’s rule, Logarithmic limits, Methods of evaluation of limits, Rationalisation method, Trigonometric limits

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