Here you'll find everything you need to know about solving limits. I prepared a list of all possible cases of limits. If you master these, there won't be a single limit you can't solve.

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.

Remember that you can **post your questions** here: Your Limits Questions. If you aren't able to solve a problem even after going through this page, you may post a question about it there.

One thing is to understand intuitively what limits are, another is to know how to solve any limit problem you might face.

To get the intuition of what limits are (I really recommend to get that before learning all the techniques presented here), you can visit the pages:

- Limits and Continuity: Here you'll find an intuitive approach to the concepts of limits and continuity. This is a good starting point.
- Limits of Functions: Here you'll go deeper into the concept of limits of functions. You'll also learn the basic properties of limits, which are used in any problem that involves solving limits.

If you haven't learned about continuous functions yet, you can skip that page.

Solving limits by factorization just requires you to remember your algebra days. 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:

* The Self-Study Course* is a complete resource that will guide you in the process of learning calculus intuitively. It is much more than a textbook, as it is specifically designed for self-study.

If you want to study more in depth what you find on this website, following the same paradigm of forming an intuitive understanding first, this is the way to go. Click here to learn more.

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