Calculus Unit Test 1A - Limits and Continuity

Front

What is a removable discontinuity?

Back

A removable discontinuity occurs at a point where a function is not defined, but can be made continuous by defining or redefining the function at that point.

Front

What is a non-removable discontinuity?

Back

A non-removable discontinuity is a point where a function is not continuous and cannot be made continuous by redefining the function at that point.

Front

Define continuity at a point.

Back

A function is continuous at a point x = c if the following three conditions are met: 1) f(c) is defined, 2) the limit of f(x) as x approaches c exists, and 3) the limit equals f(c).

Front

What does it mean if a limit approaches infinity?

Back

If a limit approaches infinity, it means that as the input values get closer to a certain point, the output values increase without bound.

Front

Calculate the limit: @@ ext{lim}_{x ightarrow-rac{1}{2}}(3x^2 + 2)@@

Back

The limit is 3(-rac{1}{2})^2 + 2 = 3(rac{1}{4}) + 2 = rac{3}{4} + 2 = rac{11}{4}.

Front

What is the limit of a constant function?

Back

The limit of a constant function as x approaches any value is simply the constant itself.

Front

What is the limit of @@ ext{lim}_{x ightarrow-rac{1}{2}}(x^2 - 4)@@?

Back

The limit is (-rac{1}{2})^2 - 4 = rac{1}{4} - 4 = -rac{15}{4}.

Front

What is the limit of @@ ext{lim}_{x ightarrow 0} rac{ ext{sin}(x)}{x}@@?

Back

The limit is 1.

Front

What is the definition of a limit?

Back

A limit is a value that a function approaches as the input approaches a certain point.

Front

What is the Squeeze Theorem?

Back

The Squeeze Theorem states that if f(x) ≤ g(x) ≤ h(x) for all x in some interval, and the limits of f(x) and h(x) as x approaches c are both L, then the limit of g(x) as x approaches c is also L.