What are shown at the bottom of the periodic table in order to keep the table a reasonable size?

Answer 1

The lanthanides are generally found in the top and bottom rows beneath the main body of the periodic table, the actinides are below them.

That's because the lantganides and actinides woukd require 14 additional columns each having only two elements apiece, meaning the table would have a lot of "dead space" above them.

Lanthanides are so called because they come after lanthanum in atomic number (lanthanum = 57, lanthanides = 58-71) and they have properties more or less similar to lanthanum. Actinides, below the lanthanides in the usual version of the periodic table, have a similar relationship with actinium.

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Answer 2

Xavier Avery

To keep the periodic table manageable in size, a group of elements known as the lanthanides and actinides are usually displayed separately at the bottom of the table.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.