LimitsDefinitionsThe limit of a function (if it exists) for some x-value, c, is the height the function gets closer and closer to as x gets closer and closer to c from the left and the right.A function has a limit for agiven x-value if the function zeroes in on some height as x gets closer and closer to the given value from the left and the right.

An example

Another example

DefinitionAnother example

Another example

Important Determining the limitNumerical Graphical Analytical (using algebra or calculus) Limits that do not exist Limits that differ from the right and from the left f(x) approaches a different number from the right side of c than it approaches from the left side of c Unbounded behaviour f(x) increases or decreases without bound Oscillating behaviour f(x) oscillates/varies between 2 fixed values as x approaches c. Limits that do not exist As x approaches 0 from the right, f(x) = 1 and as x approaches 0 from the left, f(x) = -1.

Do the following limits exist?

Do the following limits exist? More examples

Another exampleSketch the following graph:

Determine the following limits

Past Paper PracticeQuestion 5, May 2010Evaluating limits Analytical approachesPlug-and-chug (substitution) limitsAside from recall limits, plug-and-chug problems make up the other category of easy limits. Just plug the x-number into the limit function, and if the computation results in anumber, thats your answer. For example,

More on substitution Complicated limit problemsUse of algebra an example Factorisation and cancelling

Conjugation and cancelling

Simplification

Limits and infinity Limits of infinity If the highest power in the numerator is larger than the highest power in the denominator, then the limit of the function as x approaches infinity (or negative infinity) does not exist

Limits of infinity If the highest power in the numerator is smaller than the highest power in the denominator, then the limit of the function as x approaches infinity (or negative infinity) exists and equals 0.

Limits of infinity If the highest power in the numerator is equal to the highest power in the denominator, the limit is the ratio of thecoefficients of the highest power of x in the numerator and in the denominator. For example, if

then,

Past Paper PracticeQuestion 6, May 2010Assignment August 2010