2 Limits and Continuity Free Online FlashCards

What does $\lim_{x\to a}f(x)=L$ mean?

A limit describes the value that a function approaches as its input approaches a number or grows without bound.

Must $f(a)$ equal the limit?

No. A limit concerns nearby values of $f(x)$, so $f(a)$ may equal $L$, have another value, or be undefined.

Evaluate $\lim_{x\to1}\frac{x^2-1}{x-1}$.

The limit is $2$. Factoring gives $\frac{x^2-1}{x-1}=x+1$ for $x\ne1$, so the nearby values approach $2$.

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.

Evaluate $\lim_{x\to2}\frac{x^2-4}{x-2}$.

The limit is $4$. Factor $x^2-4=(x-2)(x+2)$, cancel for $x\ne2$, and then evaluate $x+2$ at $2$.

Evaluate $\lim_{x\to0}\frac{\sqrt{x+1}-1}{x}$.

The limit is $\frac{1}{2}$. Multiply by the conjugate to obtain $\frac{1}{\sqrt{x+1}+1}$, then substitute $x=0$.

Can a hole coexist with an existing limit?

A hole does not prevent a limit from existing. If both sides of the graph approach the same $y$-value, that value is the limit.

What is $\lim_{x\to0}\frac{1}{x}$?

The two-sided limit does not exist because the right-hand limit is $\infty$ while the left-hand limit is $-\infty$.

How is a horizontal asymptote found when degrees match?

For a rational function, if the numerator and denominator have equal degree, the horizontal asymptote is the ratio of their leading coefficients.

What are the conditions for continuity at $x=a$?

A function is continuous at $x=a$ when $f(a)$ is defined, the limit exists, and $\lim_{x\to a}f(x)=f(a)$.

What is a removable discontinuity?

A removable discontinuity has a limit, but the function is undefined there or has a value different from that limit.

What defines a jump discontinuity?

A jump discontinuity occurs when the left-hand and right-hand limits both exist but are unequal.