Properties of Limits Examples

Let  and . Evaluate .

Here's a limit that's impossible to find without using properties of limits. The first one we'll use if the Quotient Rule, to split the top and bottom of the fraction into their own limits.

Now we can pull the 3 out of the top using the Constant Multiple Rule.

This is now a simple plug-and-chug problem.

But can you plug, chug, and chew gum at the same time?

Use the limit properties to evaluate .

We'll start off by using the Quotient Rule to split up the fraction. Then we can start unraveling that mess up top.

Let's carry on with the Product Rule.

We can use the Constant Multiple and Power Rules on 3x².

Now that the problem is broken into three cute little, tiny, itty, bitty limits, it's solving time.

Finally, everything simplifies down to a nice, neat 12.

Evaluate  using the properties of limits.

Here we have a funky radical here. We would be completely stuck, except that we know that we can rewrite it as a fractional exponent.

We combine exponents like this by multiplying them together, and then we can use the Power Rule to whisk them all away, outside of the limit.

Now we can evaluate the limit.

And that is that. Let's not waste our time trying to reduce the radical any more, because we can't.