Calculating limits by subbing in values

[tahayassen]

Doesn't solving limits by substituting values defeat the point? For example: you can solve a limit of a quadratic by just subbing in a x value. But how do we know limit as x approaches a is the same as f(a)?

 

[haruspex]

Please clarify with an example.

 

[Hanabijm]

I'll try to take a stab at what you're getting at :)

For limits, we are concerned what y-value the function is headed for. It doesn't matter if there is a hole or something at that point, because the limit never actually equals the value x is headed towards. It can get infinitely close, however.

For some limits, we don't have to do much work or simplification. So, if you can plug in the x-value without the denominator being equal to 0, go for it!

Also, if lim x-> a of f(x) exits and equals f(a) , we call the function continuous (Assuming f(a) exists as well)

 

[tahayassen]

Never mind.

My prof. answered my question in lecture today. It's called the direct substitution property.

http://web.viu.ca/wattsv/math121/Overheads/section14/directsubstitution.pdf