How Do I Find the Limit of an Equation With a Square Root in the Denominator?

In calculus, limits are used to discuss the behavior of a function and its graph. By designating a limit at a given x-value, a mathematician can describe how that function might approach a coordinate even if the function at that value cannot produce an existing number.

  1. Determine when the function does not exist

    Using your knowledge of the rules of variables in roots and denominators, figure out at what values or ranges the y-value of a function is incalculable. You cannot take the square root of a negative number, and a fraction cannot have a 0 denominator, so you cannot plug in x-values that will create either of these conditions.

  2. Track the values as x approaches the limit

    If you are asked to find the limit when x=2 of f(x), it could be the same as the y-value. If you can plug x into the equation and get a real value, that probably is the limit. However, if the x-value would create a square root of a negative number or a 0 denominator, you need to look at how the function approaches that x value. Create a table with x-values that get really close to the coordinate indicated. In this case, you might use 1, 1.9, 1.99, 2.01, 2.1 and 3. Plug each of these into the function, and record their values. Alternatively, if you are asked to find the limit as x approaches infinity or negative infinity, pick a few very high values with increasing orders or magnitude and plug those in.

  3. Find the nearest value approached

    If the values taken at x=1.99 and x=2.01 are very close together, then you can assume the value of the limit is between them. If the values as x approaches infinity grow exponentially larger or grow closer and closer to 0, then the approached value is the limit.