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Limits Solved Examples

Last updated date: 23rd May 2024
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Limits Questions JEE

Solving Limits or any Calculus Limit requires simple mastering over the techniques given here. In this section, you will find everything you need to know about solving limits questions JEE and calculus problems involving limits. At Vedantu, the experts have prepared a list of all possible cases of problems which are an ultimate resource for solving limits. Using these techniques, you will be able to solve any kind of problem involving limits in calculus. You'll also find limits solved problems PDF and tips for every type of limit in calculus. However, if you still want to receive more lessons covering everything in calculus directly into your email, get on-board with Vedantu experts.

How To Solve Limits

Let’s get started to learn the idea behind limits and problem-solving techniques. Following are the various Limits of a Function:

  • Evaluate limits using direct substitution

  • Evaluate limits using factoring and cancelling

  • Evaluate limits by expanding and simplifying

  • Evaluate limits by combining fractions

  • Evaluate limits by multiplying by the conjugate

Type 1: Limits by Direct Substitution

These are the simplest problems involving limits in calculus. In these problems, you are only required to substitute the value to which the independent value is approaching. As a limits examples and solutions:

                       Lim x²

                                x → a

In the case, if ‘f’ is a polynomial and ‘a’ is the domain of f, then we simply replace ‘x’ by ‘a’ to obtain:-

                         Lim x²

                          x → a – a²

The technique we use here is related to the concept of continuity. You can also solve Limits by Continuity.

Type 2: Limits by Factoring

Now this one is an interesting way of solving limits. In these limits, if you try to substitute, you get an indetermination. For example:

                       Lim x² x² - 1/ x-1

                               x → 1

If you simply substitute x by 1 in the mathematical equation, you will get 0/0. So, what can be done? We can use our algebraic skills to simplify the expression. In the example given earlier, we can factor the numerator:

Lim x² x² - 1/ x-1 = lim (x-1) (x+1)/ x-1 = lim (x+1) = 2

x → 1          x → 1               x → 1

You will spot these types of problems easily whenever you observe a quotient of two polynomials. You could attempt this technique if there is an indetermination.

Type 3: Limits by Rationalization

This type of technique involves limits with square roots. In these types of limits, we use an algebraic technique called rationalization to solve limits. For example:

              Lim 1- √x / 1-x

              x → 1          

If we simply substitute, we obtain 0/0 and we cannot factor this. The strategy is to multiply and divide the fraction by an appropriate expression. (Keep in mind that if you multiply and divide a number by the same element you obtain the same number). In this case, we use the identity given below:

                   (√a - √b) (√a + √b) = a - b

You need to only perform the product on the left to verify it. So, as and when you spot the sum or difference of the two square roots, you can use the previous identity to the case. The two factors on the left are what we call conjugate expressions.

Solved Examples on How To Solve Limits

You will find the following types of limits examples and solutions in the JEE limits question bank provided by Vedantu.

Example: Identify the limit of the following expression?

Lim  x² - 5 / x² + x - 30

x → 5


Though the limit given is the ratio of two polynomials, x = 5. This makes both the numerator and denominator equal to zero (0). We have to factor both numerator and denominator as given below.

Lim (x – 5) (x + 5) / (x – 5) (x + 6)

x → 5

Simplify the expression to get:-

Lim (x + 5) / (x + 6) = 10/11

x → 5

FAQs on Limits Solved Examples

Q1. What is a Limit?

Ans. The limit is basically an idea of approaching to find a limit. Usually, there are three ways to approach to calculating limits:

  • Numerical Approach: t-table

  • Analytical Approach: use calculus or algebra

  • Graphical Approach: evaluate the graph

For a wide array of limited edition question banks and Limit examples and solutions in free PDF file format, you can seek assistance from Vedantu.

Q2. How do we express Limits?

Ans. Mathematically, we write and say “the limit of a function f(x), as x approaches a, is equivalent to L”. If we can make the values of function f(x) arbitrarily close to L by taking x to be adequately close to 'a' (on either side of a) but not equivalent to a.

This is to say that as 'x' becomes closer and closer to the number a (from either side of a), the value of f(x) gets much nearer to the number ‘L’. In computing the limit of f(x) as x approaches, remember that we never take into account x = a. 

F(x) is needless to be even defined when x = a. One factor that matters is how f(x) is defined close to 'a'. You will find a collection of Limits solved problems PDF free which will be very helpful for your board as well JEE 2021 exam preparation.

Q3. What is the Limit Theorem?

Ans. While x approaches c, the limit of the function f(x) is L. If the limit from the right exists and the limit from the left exists, both limits are L.