How do you find the limit of expressions without graphs?

[939]

Homework Statement

I cannot remember how to do this. How do you find the limit of expressions, without graphs of them?

Homework Equations

i.e.

lim x -> 3 (x^2 - 5x + 2)

The Attempt at a Solution

1) (3 + h)^2 - 5(3 + h) + 2
= (3 + h)(3 + h) - 15 - 5h + 2
9 + 6h - h^2 - 15 - 5h + 2
-4 - 1h - h^2
...
The book says it is -4, but is this the right method? Will it work for rational expressions too?

 

[Mark44]

Homework Statement

I cannot remember how to do this. How do you find the limit of expressions, without graphs of them?

Homework Equations

i.e.

lim x -> 3 (x^2 - 5x + 2)

The Attempt at a Solution

1) (3 + h)^2 - 5(3 + h) + 2
= (3 + h)(3 + h) - 15 - 5h + 2
9 + 6h - h^2 - 15 - 5h + 2
-4 - 1h - h^2
...
The book says it is -4, but is this the right method? Will it work for rational expressions too?

f(x) = x² - 5x + 2 is a polynomial, hence it is continuous everywhere. To find the limit of f as x → 3, simply evaluate f(3). You should get -4.

It looks like you are misremembering part of the difference formula in the definition of the derivative.

 

[939]

f(x) = x² - 5x + 2 is a polynomial, hence it is continuous everywhere. To find the limit of f as x → 3, simply evaluate f(3). You should get -4.

It looks like you are misremembering part of the difference formula in the definition of the derivative.

Thanks, got it.

In the case of a rational expression, I merely:

1) Evaluate at f(#)
2) If that = 0, try to see if it can be factored and then evaluate again

Correct?

 

[Dick]

Thanks, got it.

In the case of a rational expression, I merely:

1) Evaluate at f(#)
2) If that = 0, try to see if it can be factored and then evaluate again

Correct?

Yes. But if f(x)=(x-1)/(x^2-1) and you want the limit as x->1, you don't get 0. You get 0/0. That's the danger sign. But you've got the idea.