Trigonometric and Polar Functions

4π

2π

π

π/2

The left and right limits exist and are finite but unequal at that point.

There are no limits because the function becomes non-real at that point.

The left and right limits agree yet differ from the function’s value at that point.

Both one-sided limits tend towards infinity but in opposite directions.

$\tan(\tan^{-1}(t))$

$\cos(\cos^{-1}(t))$

$\sec(\sec^{-1}(t))$

$\sin(\sin^{-1}(t))$

Sine wave (e.g., $h(t)=5\sin(\theta)$)

Polynomial (e.g., $f(x)=x^2$)

Tangent (e.g., $g(t)=\tan(0)$)

Quadratic (e.g., $f(t)=\sin t+5$)

It shifts the graph horizontally right/left depending on the sign of $c$.

It flattens or compresses the vertical stretch, thus making the waves appear less tall.

It heightens the entire wave vertically, increasing the distance between the midline and maximum/minimum values.

It zooms in or out and changes both the amplitude and period simultaneously.

-5

5

-3

3

Degrees

Cubic centimeters

Meters

Radians

$g(T+t)/4$

$g(T-t)$

$g(t+T)$

$g(t-\frac{T}{4})$

The limit exists but does not equal $f(c)$.

The two one-sided limits are different as $x$ approaches $c$.

The limit as $x$ approaches $c$ is infinity.

The limit does not exist as $x$ approaches $c$.

j(r) = - ext{cos}(r/-)

j(r) = ext{cos}(-r)

j(r) = - ext{cos}(-r)

j(r) = ext{cos}(r)