Djedji AB Calculus Limits

One sided Limits

lim f(x)

x→?^-

^or

lim f(x)

x→?⁺

lim f(x)

x→?^-

limit when x approaches ? from the smaller numbers / left of the number line

lim f(x)

x→?⁺

limit when x approaches ? from the larger numbers / right of the number line

Two Sided Limits

lim f(x)

x→?

lim f(x)

x→?

limit when x approaches ?⁺ and ?^- is the same

Plugging in Limits

plug in numbers for x that are slightly above and below ? and look for a pattern, if they both approach the same number, that is the limit

Infinite Limits

Positive Infinity: The function's output grows infinitely large as x approaches a value

Negative Infinity: The function's output grows infinitely small as x approaches a value

Means the Limit DNE but expressing it as infinite is more accurate

Limite DNE if

1. lim f(x) /=/ lim f(x)

____x→ ?^- ___x→ ?⁺

2. lim f(x) = ± infinity

____x→ ?

3. f(x) oscillates between fixed values of x

example: 1, -1, 1, -1, 1, -1 …

Properties of Limits

Constant Function

Identity Function

Power Function

Root Function

Constant Function

if f(x) = b

then

lim b = b

x→ ?

Identity Function

if f(x) = x

then

lim x = ?

x→ ?

Power Function

if f(x) = x^n

then

lim x^n = ?^n

x→ ?

Root Function

if f(x) = ⁿ√x

then

lim ⁿ√x = ⁿ√?

x→ ?

it doesn’t work if n is even and x<0

Operations with Limits

assume

lim f(x) = L

x→ ?

lim g(x) = K

x→ ?

Constant Multiple Rule

lim b*f(x) = bL

x→ ?

Sum and Difference Rule

lim f(x) ± g(x) = L ± K

x→ ?

Product Rule

lim f(x)*g(x) = L*K

x→ ?

Quotient Rule

lim f(x)/ g(x) = L/ K

x→ ?

if

lim g(x) /=/ 0

x→ ?

Techniques to calculate limits

1. direct substitution

2. factor and divide out

3. rationalize

4. simplify complex fractions

5. one sided limits (Piecewise Function)

6. squeeze theorem

7. trigonometric theorems

Simplify a sum of cubes

A³+B³=(A+B)(A²-AB+B²)

A³-B³=(A-B)(A²+AB+B²)

Use SOAP (Same, Opposite, Always Positive)

Synthetic Division

One Sided Limit Piecewise Function

Squeeze Theorem

if

lim g(x) = L

x→ ?

and

lim h(x) = L

x→ ?

and

g(x) <= f(x) <= h(x)

then

lim f(x) = L

x→ ?

Trigonometric Theorems

lim sin(x)/x = 1

x→ 0

and

lim sin(ax)/ax = 1

x→ 0

and

lim 1-cos(x)/x = 0

x→ 0

Limits of rational functions at infinity

if

f(x) = a_nxⁿ+ … + a₁*x+a₀ / b_m*x^m+ … + b₁*x+b₀

then

lim f(x) = ____lim a_n*xⁿ / b_m*x^m

x → ± infinity _x → ± infinity

3 cases

if

n>m

then

im f(x) = ____ ± infinity

x → ± infinity

if

n = m

then

im f(x) = ____ a_n/b_m

x → ± infinity

if

n<m

then

im f(x) = ____ 0

x → ± infinity

Limits of irrational functions at infinity

1. √x² = |x| = -x (x<0) (negative) or x ( x>= 0) (positive)

2. if x → -infinity then √x² = -x

3. if x → infinity then √x² = x

Justify End Behaviour

look at the leading term, find if its degree is positive or negative, and see if the leading coefficient is positive or negative

example: degree is even, leading coefficient is negative

goes down to negative infinity as x approaches infinity and negative infinity