After problems that help you practice limit algebra you will find problems on indeterminate ratios, indeterminate products and indeterminate powers. Other supplementary methods to practice are factoring out/cancelling, difference of roots, problems that require squeezing and problems featuring oscillation.
If you want to refer to sections of Methods Survey on limits while working the exercises, you can click here and it will appear in a separate full-size window. Similarly, here we offer Theory - Limits.
Use the limit algebra to evaluate the following limits.
Check that the type of the given problem is the indeterminate ratio, then use the L'Hospital rule to find the limit.
Check that type of the given problem is the indeterminate product. Then transform the product into a fraction. State the resulting limit that has to be found and find its type.
Check that the type of the given problem is the indeterminate power. Then use the recommended transformation to change the power into a fraction. State the resulting limit that has to be found and find its type.
In the following problems, find the limit using suitable cancelling.
In the following limits, identify the dominant term (see intuitive evaluation) and find the limit by factoring it out. Note that factoring out works thanks to the fact that dominant terms are always unique here.
In the following limit problems, identify the dominant terms in the numerator and the denominator. Then use factoring out to find the limit; in some cases one can also use cancelling.
In the following limit, identify the dominant term in the root and determine what type of expression the root is. Then factor out the dominant term and find the limit. Try also cancelling (extending the root to the whole fraction).
Use algebra to get rid of the difference of roots.
Set up the Squeeze theorem for the following limits (see comparison and oscillation).
Sometimes we conclude that a given expression oscillates with undiminished amplitude as x approaches given a, which indicates that its limit does not exists. To prove it formally we use the Heine theorem, namely we construct two sequences of values of x so that one of them highlights the haights and the other lows in the oscillation. If the given limit existed, these two sequences should lead to the same limit value of our expression, but they will not.