2. Limits Free Online FlashCards

What does a limit describe?

A limit is the value a function approaches as its input approaches a specified number, whether or not the function is defined at that input.

Evaluate lim x→1 (x² − 1)/(x − 1).

The limit is 2. For x ≠ 1, (x² − 1)/(x − 1) simplifies to x + 1, whose value approaches 2 as x approaches 1.

When does a two-sided limit exist?

A two-sided limit exists exactly when the left-hand and right-hand limits both exist and are equal.

Why does the jump function lack a limit at x = 2?

For the jump function, the left-hand limit at 2 is 0 and the right-hand limit is 3; therefore, the two-sided limit at 2 does not exist.

What do ε and δ represent in the limit definition?

ε is the allowed output error, while δ is the required input distance from a. The definition requires 0 < |x − a| < δ to imply |f(x) − L| < ε.

What δ proves lim x→3 (2x + 1) = 7?

For lim x→3 (2x + 1) = 7, choose δ = ε/2. Then |(2x + 1) − 7| = 2|x − 3| < 2δ = ε.

What condition is required for the quotient law?

The quotient law gives lim f(x)/g(x) = L/M when lim f(x) = L, lim g(x) = M, and M ≠ 0.

Evaluate lim x→3 (x² − 9)/(x − 3).

The limit is 6. Factor x² − 9 as (x − 3)(x + 3), cancel x − 3 for x ≠ 3, and evaluate the resulting expression x + 3 at 3.

Evaluate lim x→0 (√(x + 1) − 1)/x.

The limit is 1/2. Multiplying by the conjugate simplifies the expression to 1/(√(x + 1) + 1), which approaches 1/2 as x approaches 0.

What conditions permit the Squeeze Theorem?

The Squeeze Theorem applies when g(x) ≤ f(x) ≤ h(x) near a, and both bounding functions approach the same limit L.

Evaluate lim x→0 x sin(1/x).

The limit is 0. Since −|x| ≤ x sin(1/x) ≤ |x| and both bounds approach 0 as x approaches 0, the Squeeze Theorem applies.

Why is x = −3 a vertical asymptote of 1/(x + 3)²?

The line x = −3 is a vertical asymptote because both one-sided limits of 1/(x + 3)² approach +∞.