Foundations of Limits in Calculus BC

Limit

The value that a function approaches as the input approaches a specified value.

Notation for Limits

lim_{x \to c} f(x) = L, representing the limit of f(x) as x approaches c.

Limit Value (L)

The value the graph approaches.

Function Value (f(c))

The actual value of the function at x=c.

Left-Hand Limit (LHL)

lim_{x \to c^-} f(x), approaching c from values smaller than c.

Right-Hand Limit (RHL)

lim_{x \to c^+} f(x), approaching c from values larger than c.

Existence Rule for Limits

lim_{x \to c} f(x) = L if and only if LHL = RHL = L.

Limit Does Not Exist (DNE)

Occurs when left and right limits approach different values or diverge to infinity.

Graphical Estimation of Limits

Using a graph to estimate limits by observing the behavior near a target x-value.

Removable Discontinuity

A hole in the graph where the limit exists even though the function value does not.

Jump Discontinuity

When the left and right limits approach different y-values, resulting in limit DNE.

Vertical Asymptotes

Lines where a function approaches infinity or negative infinity, leading to DNE.

Basic Limit Property: Sum/Difference

lim [f(x)  g(x)] = L  M.

Basic Limit Property: Product

lim [f(x)  g(x)] = L  M.

Basic Limit Property: Quotient

lim [\frac{f(x)}{g(x)}] = \frac{L}{M} provided M  0.

Basic Limit Property: Power

lim [f(x)]^n = L^n.

Direct Substitution Strategy

Plugging in c to check the limit; if you get a real number, that's the limit.

Indeterminate Form (0/0)

Occurs when direct substitution results in 0/0, indicating further work is needed.

Factoring and Canceling Technique

A method to solve limits for polynomial rational functions by removing conflicting terms.

Rationalizing with Conjugates Technique

Multiplying by the conjugate to eliminate square roots in limit evaluations.

Special Trigonometric Limits

Limits like
lim_{x \to 0} \frac{\sin x}{x} = 1 are essential for evaluating certain limits.

Squeeze Theorem

If g(x) (x)  h(x) and both g and h approach L, then f also approaches L.

Common Mistake: Confusing f(c) with the Limit

Just because f(c) is undefined does not mean the limit does not exist.

Common Mistake: Misinterpreting 0/0

0/0 is indeterminate and requires further analysis, not a direct DNE.

Common Mistake: Neglecting One-Sided Checks

For piecewise functions, always check both sides to ensure limit existence.

Common Mistake: Notation Errors

Keep using
lim notation until substituting the actual value.