AP Calculus - Beginning Limits

End Behavior

100

The end behavior model for -3x² + 4x +20

-3x²

200

The right end behavior model for x + sin(x)e^-5x

x

300

The end behavior model for (2x⁵ + x⁴ - x²)/(3x² + 7x + 7)

2x³/3

400

The horizontal asymptote for (3x+6x⁴-9x²+4)/(11+2x-5x⁴)

y = -6/5

500

A function g is a left end behavior model for f if and only if (iff)...

lim_{(x->neg inf)} f(x)/g(x) = 1

Unit Circle and Squeeze Theorem

100

The bounding function for x⁴sin(3/x)

x⁴

200

What is the trig value that corresponds to the x-coordinate?

cos(theta)

300

The inequality I start with when using squeeze theorem for x²cos(1/x²)

-1 ≤ cos(1/x²) ≤ 1

400

The tangent of what angle is 1?

Pi/4 or 5Pi/4

500

The sine of what angle in the fourth quadrant is -1/2

11Pi/6

Limits

100

lim_(x→8)⁡ 7+ √(x+1)

10

(direct substitution)

200

lim_(x→4)⁡ (x²-5x+4)/(x²-2x-8)

1/2

(factor)

300

lim_(x→9)⁡ (x - 9) / (√x - 3)

6

(factor or multiply by conjugate)

400

lim_(x→0)⁡ [1/(x+3) - 1/3] / x

-1/9

(simplify with LCD for fraction)

500

lim_(x→1) [√(x+3) - 2] / (x-1)

1/4

(multiply the conjugate)

Continuity and Limit Knowledge

100

These three things must be equal for f to be continuous at x=2

lim_(x→2+)⁡ = lim_(x→2-)⁡ = f(2)

200

If the limit of f(x) as x approaches 2 is 4, what can you conclude about f(2)?

Nothing

300

The 4 types of discontinuity

Removable, jump, infinite, oscillating

400

The 6 properties (or rules) of limits

Sum, Difference, Product, Constant Multiple, Quotient, Power

500

Name three times when a limit does not exist.

1. f(x) approaches a different number from left and right side of c 2. f(x) increases or decreases without bound 3. f(x) oscillates between two fixed values

Limits of Trig Functions

100

lim_(x→0) ⁡cos⁡x

1

300

lim_(x→0)⁡ cos⁡(x)tan(x)/x

1

400

lim_(x→π)⁡ (x)sec(⁡x)

-π

500

lim_(x→π/2)⁡ cos(x) / (x-π/2)

-1

Use trig identity that cos(x) = sin(π/2 - x)

Should know lim_(x→0)⁡ sin(x) / x = 1