Limits

lim(x🡒0)[f(x)] = 1

The limit as x approaches 0 is 1.

Limit is what y value a function approaches at x, not what y is at x.

A limit can exist even if there's a hole in the graph.

Infinite Limits

lim(x🡒a)[f(x)] = ±∞

Limits at Infinity

lim(x🡒±∞)[f(x)] = a

One-Sided Limits

Two sided limits only exist if the left and right hand limits both match.

Left Hand Limits

lim(x🡒0⁻)[f(x)] = 1

The limit as x approaches 0 from the left is 1.

Right Hand Limits

lim(x🡒0⁺)[f(x)] = 1

The limit as x approaches 0 from the right is 1.

Three Ways to Solve Limits

Factor and Cancel

Usually a polynomial over a polynomial (a rational function).

Conjugate

X is generally trapped under a square root.

Multiply the top and bottom by the exact same thing on the bottom, except you change the sign between the two terms.

Bottom multiplies into a difference of squares.

• The middle term is going to cancel, and you're left with the first term squared, minus the second term squared.

Multiply by the common denominator of the small fraction

Useful when you have fractions within fractions

You can then combine this with the other methods after simplifying the fractions.

Trigonometric Limits

lim(x🡒0)[sin(x)/x] = 1

A Special Trigonometric Limit

Limit Laws

The limit of a constant function is the constant itself.

The Limit of a Constant (Horizontal Line) Function

The limit of an identity function as x -> a is a.

The Limit of an Identity (Sloped Line) Function

The limit of a function multiplied by a constant is the constant multiplied by what the limit is approaching.

The Limit of a Constant Multiple of a Function

The limit of a function taken to a power is the limit value also taken to that power.
If the limit value not-equal to 0, then the limit of a function taken to a negative power is the limit value also taken to that negative power, or 1 divided by the limit value taken to that power.

Limits of Integral Powers

The the function is non-negative, then the limit of a function under the root of a power is the limit value also taken to the root of that power.

Limit of the Root of a Function

The limit of a function to a rational power is the limit value also taken to the same rational power.

Limit of a Function to a Rational Power

The limit of two functions composed is the composition of the inner function's limit into the outer function.

Limit of a Composition of Functions

Arithmetic Limits

Sum Rule

The limit of two functions added together is the sum of the limits of each function at a.

Difference Rule

The limit of two functions subtracted is the difference of their two limits at a.

Product Rule

The limit of two functions multiplied is the product of their two limits at a.

Quotient Rule

The limit of two functions divided is the quotient of their two limits at a.