Solve Any Calculus Limit

Here you'll find everything you need to know about solving calculus problems involving limits. I prepared a list of all possible cases of problems. If you master these techniques, you will be able to solve any type of problem involving limits in calculus.

My goal for this page is to be the ultimate resource for solving limits. **It may be a good idea to bookmark it**. You'll find solved examples and tips for every type of limit.

Here we focus on problem-solving techniques. If you want to get the intuition behind the idea of limits, please visit these pages:

- Limits and Continuity: Intuitive Introduction
- Limits of Functions: More Intuition and Simple Examples

These are easiest problems. In these problems you only need to substitute the value to which the independent value is approaching. For example:

Here we simply replace x by a to get

I don't think you need much practice solving these. They're not much fun either. However, there is an interesting question here by a reader that relates the technique we use here and the concept of continuity: Solving Limits by Continuity.

Now this is more interesting. In these limits, if you try to substitute as in the previous case, you get an indetermination. For example:

If you simply substitute x by 1 in the expression you'll get 0/0. So, what can we do? We use our algebraic skills to simplify the expression. In the previous example we can factor the numerator:

It is easy to spot this type of problems: whenever you see a quotient of two polynomials, you may try this technique if there is an indetermination.

Watch this video for more examples:

In these limits we apply an algebraic technique called rationalization. For example:

In the example above, the conjugate of the numerator is:

All you need to do is to multiply and divide by the conjugate of the numerator and work algebraically.

Here's another worked out example: Limit by Rationalization. There are other examples that are trickier, in the sense that you need to multiply by two expressions. For example:

In this case you need to multiply and divide by two factors: the conjugate of the numerator and then the conjugate of the denominator.

This problem is good practice and I recommend you to try it. If you tried and still can't solve it, you can post a question about it together with your work.

With things involving trigonometric functions you always need practice, because there are so many trigonometric identities to choose from.

In the following page you'll find everything you need to know about trigonometric limits, including many examples: The Squeeze Theorem and Limits With Trigonometric Functions.

Here also more examples of trigonometric limits. I think you'll find all techniques you need to know in these:

Here you can find a more elaborate example: Limit at Infinity Involving Number e.

This rule says that to find the limit of a quotient, you only need to find the derivatives of both the numerator and denominator and apply the limit again.

This works only if the quotient is an indeterminate form 0/0 or infinity over infinity. For example:

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