Two-Sided Limits, Squeeze Theorem, and L'Hopital's Rule

Can you skip steps if the function is always continuous?

Yes

Can you skip steps if the function has a removable discontinuity at the limit?

Yes

Can you skip steps if the function is piecewise and non-continuous at the limit (but is not a removable discontinuity)?

No

sin(x)/x

1

(1-cos(x))/x

0

What does the Squeeze Theorem state?

If f(x) and g(x) are squeezed by h(x) at a point and both approach the same limit 'L' at that point, then h(x) also approaches 'L'.

Which limits do you need to know to use the Squeeze Theorem?

Both outer functions

Is 1/1 an indeterminate form?

No

Is 0/0 an indeterminate form?

Yes

Is ∞/∞ an indeterminate form?

Yes

Is 0/∞ an indeterminate form?

No

What rule must be used when the normal finding of a limit produces an indeterminate form?

L’Hospital’s Rule

What do you take the derivative of in L’Hospital’s Rule?

Both the numerator and denominator (separately)

What needs to be written before using L’Hospital’s Rule?

“This limit produces the indeterminate form…”

Can you use L’Hospital’s Rule multiple times as long as you want?

No

Can you use L’Hospital’s Rule multiple times until the indeterminate form is gone?

Yes

Can you use L’Hospital’s Rule without an indeterminate form?

No

What is a good method for finding limits with radicals?

Rationalize denominator using conjugate pair

What is a good method for finding limits with polynomials?

Factoring

What is a good method for finding limits with a squared trigonometric function?

Replace it using cos²(x)+sin²(x)=1