Calculus Limits to Infinity Flashcards

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.