Calculus Limits to Infinity Flashcards

Questions and Answers

If the highest power of x is in the numerator, then the limit will equal...

Infinity or negative Infinity (correct)

Zero

The ratio of the coefficients

That number

If the highest power of x is in the denominator, what will the limit equal?

zero

If the highest power of x is in both the numerator and denominator, what does the limit equal?

the ratio of the coefficients of those terms

If f(c) equals a number, what will the limit equal?

that number Signup and view all the answers

If f(c) equals a number/0 where the number is not 0, what can the limit be?

Negative infinity Signup and view all the answers

What must you do if f(c) is 0/0?

perform some algebraic technique to simplify the function and then plug c back in Signup and view all the answers

You can use direct substitution for limits of piecewise functions.

False Signup and view all the answers

After using direct substitution, if the answer is a real number for absolute value functions, what is the limit?

that number Signup and view all the answers

If after direct substitution, the answer is 0/0 for absolute value functions, what should you check?

both one-sided limits as x goes to c Signup and view all the answers

You can use direct substitution for limits of greatest integer functions.

False Signup and view all the answers

Study Notes

Limits to Infinity

When the highest power of x is in the numerator, the limit approaches either infinity or negative infinity, influenced by the leading coefficient and the specific conditions of the limit.
If the highest power of x is in the denominator, the limit equals zero.
With the highest power of x in both the numerator and denominator, the limit is determined by the ratio of the coefficients of those terms.

Limits to a Number C

When f(c) gives a specific number, the limit equals that number.
If f(c) results in a non-zero number over zero, the limit may be infinity or negative infinity depending on one-sided limits. If both one-sided limits are positive, the limit is infinity; if both are negative, it is negative infinity.
When the one-sided limits diverge, the limit does not exist (DNE).
For an indeterminate form of 0/0, apply algebraic techniques (like factoring) to simplify the function before reevaluating the limit by substituting c.

Limits of Piecewise Functions

Direct substitution is not applicable; both one-sided limits must be assessed as x approaches c.
The left-hand limit utilizes the function for x < c and the right-hand limit for x > c.
If the one-sided limits are equal, that value is the limit; if they differ, the limit DNE.

Limits of Absolute Value Functions

After direct substitution results in a real number, the limit is that number.
Upon obtaining 0/0, both one-sided limits should be checked. If approaching c from the positive side, remove absolute value bars; if from the negative side, replace the inside of the absolute value with its negative before solving.
The limit exists if both one-sided limits are equal; if not, it DNE.

Limits of Greatest Integer Functions

Direct substitution cannot be used; instead, assess both one-sided limits as x approaches c.
The right-hand limit of [x] approaches c; the left-hand limit approaches the greatest integer less than c.
Equality of the one-sided limits determines the overall limit; differing results indicate that the limit DNE.

Description

This quiz focuses on understanding limits to infinity in calculus. Each flashcard presents a key concept related to limits involving the highest power of x in either the numerator or denominator. Test your knowledge and improve your calculus skills with these essential definitions.