Limits are among the most intuitive concepts in calculus, and through them, we can understand how functions relate to specific points. It allows us to understand the behavior of functions at or as the input values approach specific points or become infinitely large. Many of the algebraic functions we might form using polynomials and rational expressions cannot be carefully analyzed without understanding limits.
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In this article, we shall discuss the concept of Limits of Algebraic Functions. This falls under the broader category of Calculus, which is one significant chapter within Class 11 Mathematics. It is essential not only for board exams but also for competitive exams, including Joint Entrance Examination Main and other entrance exams, namely: SRM Joint Engineering Entrance, BITSAT, WBJEE, and BCECE. A total of five questions have been asked on this topic in JEE Main from 2013 to 2023 including one in 2017, one in 2019, one in 2020, one in 2021, and one in 2022.
A limit is defined as a value that a function approaches using the given input values. In other words, the limiting value of a function
that is to say when
The determination of limits of algebraic functions can be done in several ways: substitution, factoring, rationalizing, and using particular theorems about limits. Let's now discuss each one:
To find
- If
- If
- If
(i)
(ii)
(iii)
In this method, we factorize numerators and denominators. The common factors are canceled out and the rest of the output is the final answer.
Illustration 1:
Evaluate
we can re - write,
This method is used when either numerator or denominator or both have fractional powers (like
Let’s go through an illustration to understand better
Illustration 2:
Evaluate
Rationalizing numerator and denominator we get,
Certain limits are evaluated using known results, such as:
Example 1: Let
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1)
2)
3)
4)
Solution:
Maxima of
Minima of
By method of factorisation,
Hence, the answer is the option 1.
Example 2: The value of
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1)
2)
3)
4)
Solution:
Hence, the answer is the option 3 .
Example
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1) $36$
2)
3)
4) None of these
Solution:
Rationalizing numerator and denominator we get,
Rationalizing numerator and denominator we get,
Multiply by the conjugate of
Hence, the answer is the option (1).
Example 4:
1)
2)
3)
Solutions:
Here we can apply the concept of Limit using the rationalization method, discussed above with an example.
The limit is of
Hence, the answer is the option (2).
Example 5: Example 5: Let
Then
1)
2)
3)
4)
Solution:
For limit to exist
Hence the answer is the option (1).
Evaluating the value of a function is impossible without the application of limits. The concept of limit is the cornerstone on which the development of calculus rests. Calculus has many applications in various domains like physics, biology, engineering, etc.
To evaluate the limit of an algebraic function using direct substitution, you simply substitute the value of the variable into the function. If the function is continuous at that point, the limit is equal to the function's value at that point. For example,
If direct substitution results in an indeterminate form like
For limits involving square roots or other irrational expressions, rationalizing the numerator or the denominator can be helpful. This involves multiplying by a conjugate to simplify the expression. For example,
Limits help in understanding the behavior of functions as they approach infinity or negative infinity. This is crucial for analyzing the long-term behavior of functions. For example, the limit
A limit is defined as a value that a function approaches using the given input values.
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