How to Find the Limit of a Rational Function with Square Roots Expressions in the Numerator

Since this article will be using square root symbols, which is difficult to express in the text section of the steps below, most of the work done with the function will be shown in the images.

Things You'll Need

• Paper
• and Pencil

Instructions

1. 

   Please see the image for the function we will be working with. In order to solve a Limit Function, we must first use Direct Substitution. To do this, replace x in the function with the number that x is approaching, which in this case, is 0. If the result of this is a real number, then the limit is that number. If it yeilds 0/0, which is meaningless, as it does with this function, then there is more work to be done. Please click on the image for a better understanding.

2. 

   Since Direct Substitution resulted in 0/0, we must rewrite the function so that it is exactly the same as the orginal function except at the point x=0. Since there are square roots in the numerator of the fraction, we must multiply the numerator and the denominator by the opposite operation (conjugate) of the square roots in the numerator. Please click on the image for a better understanding.

3. 

   Now, we must simplify. A square root multiplied by the same square root is the number or function that is inside the radical sign, called the radicand. Next, we combine like terms, and then divide out x. This will give us our new, rewritten, function. Please click on the image for a better understanding.

4. 

   Since the new function is exactly the same as the original function except at x=0, we can now try Direct Substituion again. Since square roots are preferred to be in the numerator of a fraction, known as rationalizing the denominator, multiply both sides of the fraction by the square root that is in the denominator. The square root will now be in the numerator. This will give us the final answer to the limit function. Please click on the image for a better understanding.