The Limits of a Function



In this lesson, we will learn

  • the definition of limits
  • how to evaluate limits using direct substitution
  • how to evaluate limits using factoring and cancelling
  • how to evaluate limits by combining fractions
  • how to evaluate limits by multiplying by the conjugate
  • how to evaluate limits by expanding and simplifying

We have also included a limits calculator at the end of this lesson. This math tool will show you the steps to find the limits of a given function.

Related Topics: More Lessons on Calculus

Definition of Limits

We write

and say “the limit of f(x), as x approaches a, equals L

if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a (on either side of a) but not equal to a.

This says that as x gets closer and closer to the number a (from either side of a) the values of f(x) get closer and closer to the number L In finding the limit of f(x) as x approaches, we never consider x = a. In fact, f(x) need not even be defined when x = a. The only thing that matters is how f(x) is defined near a.

What is a Limit? Basic Idea of Limits
Basic Idea of Limits and what it means to calculate a limit.



Direct Substitution Property

If f is a polynomial or a rational function and a is the domain of f, then

 

Example:

Evaluate the following limits

 

Solution:

How to calculate the limit of a function using substitution.

Functions with Direct Substitution Property are called continuous at a. However, not all limits can be evaluated by direct substitution. The following are some other techniques that can be used.

Factoring and Cancelling

Example:

Solution:

We can’t find the limit by substituting x = 1 because

is undefined


Instead, we need to do some preliminary algebra. We factor the numerator as a difference of squares and then cancel out the common term (x – 1)

Therefore,


Note: In the above example, we were able to compute the limit by replacing the function by a simpler function g(x) = x + 1, with the same limit. This is valid because f(x) = g(x) except when x = 1.

Calculating a Limit By Factoring and Cancelling



How to calculate the limit of a function by using the factorisation method



If there are fractions within fractions, try to combine the fractions

Example:


Solution:

We cannot use the substitution method because the numerator and denominator would be zero.

Calculating a Limit by Getting a Common Denominator



If there is a square root, try to multiply by the conjugate

Example:

Solution:

We cannot use the substitution method because the numerator and denominator would be zero.

Calculating a Limit by Multiplying by a Conjugate





Calculating a Limit by Expanding and Simplifying







A Limits Calculator or math tool that will show the steps to work out the limits of a given function. Use it to check your answers.




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