Calculus: Limits and Indeterminate Forms

Questions and Answers

What does BOBO BOTN EATS DC stand for?

Bigger on Bottom: 0 (Zero)

Bigger on Top: None

Exponents Are The Same: Divide Coefficients

All of the above (correct)

Why is BOTN different for limits?

The 'None' applies to horizontal asymptotes, not limits. If there is no horizontal asymptote, then the limit is either positive or negative infinity.

What is the order, from smallest to largest, of terms in a fraction when the limit goes to positive or negative infinity?

Constant < log < √ (roots) < x², x³ (powers) < 2ⁿ, eⁿ (exponentials)

What is the limit when a number is raised to the negative infinity?

Zero Signup and view all the answers

What are the only two indeterminate forms that you can use L'Hospital's Rule with?

0/0 and ±∞/±∞ Signup and view all the answers

What are the other indeterminate forms?

0 times ∞, ∞-∞, 0⁰, ∞⁰, and 1^∞ (1 to the power of infinity) Signup and view all the answers

What do you do to find the limit when direct substitution results in 0 times ∞?

Divide by the reciprocal of one of the factors. Signup and view all the answers

What do you do to find the limit when direct substitution results in ∞-∞?

Find a common denominator and keep in mind that infinity minus infinity doesn't equal zero. Signup and view all the answers

What do you do to find the limit when direct substitution results in 0⁰, ∞⁰, or 1^∞?

Use log, set to y= and take ln of both sides. Signup and view all the answers

What does ln(1)^∞ equal?

∞ times 0 Signup and view all the answers

What does ln(0)⁰ equal?

0 times ∞ Signup and view all the answers

What does ln(∞)⁰ equal?

0 times ∞ Signup and view all the answers

What does 0^∞ equal?

0 Signup and view all the answers

Study Notes

Limits and Indeterminate Forms

Limits approach values as ( x ) goes to positive or negative infinity.
The acronym BOBO BOTN EATS DC assists in determining limits at infinity.

Interpretation of BOBO BOTN EATS DC

"Bigger on Bottom" leads to a limit of zero.
"Bigger on Top" signifies no limit exists.
"Exponents Are The Same" necessitates dividing coefficients to find the limit.

Horizontal Asymptotes and Limits

"None" pertains specifically to horizontal asymptotes and not limits; limits may diverge to positive or negative infinity if no asymptote exists.

Order of Growth of Terms

The growth order from smallest to largest when evaluating limits is:
- Constant < log < √ (roots) < ( x^2 ), ( x^3 ) (powers) < ( 2^n ), ( e^n ) (exponentials).

Limits involving Negative Infinity

The limit of a number raised to negative infinity is always zero.

Indeterminate Forms

L'Hospital's Rule is applicable only for forms ( 0/0 ) and ( ±∞/±∞ ).
Other indeterminate forms include ( 0 \times ∞ ), ( ∞ - ∞ ), ( 0^0 ), ( ∞^0 ), and ( 1^∞ ).

Finding Limits in Indeterminate Forms

For ( 0 \times ∞ ), convert to a limit by dividing by the reciprocal, e.g., ( x \ln(x) ) becomes ( \ln(x)/(1/x) ).
For ( ∞ - ∞ ), establish a common denominator and apply log properties where necessary; note that infinity minus infinity does not yield zero.
For ( 0^0 ), ( ∞^0 ), or ( 1^∞ ), take natural logarithm, manipulate, and then exponentiate to recover the final limit.

Specific Limit Evaluations

( \ln(1)^{∞} ) resolves to ( ∞ \ln(1) ) which equals zero.
( \ln(0)^{0} ) evaluates to ( 0 \times ∞ ), resulting in an indeterminate form.
( \ln(∞)^{0} ) also leads to ( 0 \times ∞ ), implying a need for careful evaluation.
The limit ( 0^{∞} ) is equal to zero.

Description

This quiz covers the fundamental concepts of limits and indeterminate forms in calculus. Topics include evaluating limits at infinity using the BOBO BOTN EATS DC acronym, understanding horizontal asymptotes, and applying L'Hospital's Rule. Test your knowledge of the order of growth of terms and how they affect limits.