What are the differences between direct substitution and algebraic manipulation for finding limits?
Direct Substitution: Plug in the value directly. | Algebraic Manipulation: Simplify the expression before plugging in the value (factoring, conjugates, etc.).
What are the differences between using a graph and a table to find limits?
Graph: Visual representation of the function's behavior. | Table: Numerical representation of the function's values.
What are the differences between L'Hôpital's Rule and the Squeeze Theorem?
L'Hôpital's Rule: Used for indeterminate forms by taking derivatives. | Squeeze Theorem: Used to find limits by bounding a function between two others.
What are the differences between one-sided limits and two-sided limits?
One-sided limits: Approach from either the left or the right. | Two-sided limits: Approach from both sides, and both must agree for the limit to exist.
What are the differences between a limit existing and a function being continuous at a point?
Limit exists: The function approaches a value. | Continuous: The limit exists, the function is defined, and they are equal.
What are the differences between removable and non-removable discontinuities?
Removable: Can be 'fixed' by redefining the function at a single point. | Non-removable: Cannot be fixed (e.g., jump, asymptote).
What are the differences between finding limits at finite values and finding limits at infinity?
Finite values: Focus on the function's behavior near a specific point. | Infinity: Focus on the function's end behavior as x grows without bound.
What are the differences between using limit laws and algebraic manipulation?
Limit Laws: Apply basic properties of limits (sum, product, etc.). | Algebraic Manipulation: Transform the expression to simplify it before applying limit laws.
What are the differences between indeterminate forms 0/0 and ∞/∞?
0/0: Both numerator and denominator approach zero. | ∞/∞: Both numerator and denominator approach infinity.
What are the differences between using conjugates and factoring when simplifying limits?
Conjugates: Used for expressions with radicals. | Factoring: Used for polynomial expressions.
Sum Limit Law Formula
$\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$
Difference Limit Law Formula
$\lim_{x \to c} [f(x) - g(x)] = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)$
Product Limit Law Formula
$\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$
Quotient Limit Law Formula
$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$, provided $\lim_{x \to c} g(x) \neq 0$
L'Hôpital's Rule Formula
If $\lim_{x \to c} f(x) = 0$ and $\lim_{x \to c} g(x) = 0$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$
Constant Multiple Limit Law
$\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)$
Power Limit Law
$\lim_{x \to c} [f(x)]^n = [\lim_{x \to c} f(x)]^n$
Limit of a Constant
$\lim_{x \to c} k = k$
Limit of x
$\lim_{x \to c} x = c$
Squeeze Theorem Formula
If $f(x) \leq g(x) \leq h(x)$ for all x near c, and $\lim_{x \to c} f(x) = L = \lim_{x \to c} h(x)$, then $\lim_{x \to c} g(x) = L$.
How does a hole in a graph relate to limits?
A hole indicates that the function is not defined at that x-value, but the limit may still exist if the function approaches a specific y-value.
How does a vertical asymptote on a graph relate to limits?
A vertical asymptote indicates that the limit approaches infinity (or negative infinity) as x approaches a certain value from one or both sides.
What does it mean if a graph oscillates rapidly near a point when considering limits?
It suggests that the limit does not exist because the function does not approach a single value.
How can you identify one-sided limits from a graph?
Examine the behavior of the function as x approaches the value from the left and from the right separately.
How does the graph of a function indicate continuity at a point?
The graph must not have any breaks, holes, or jumps at that point. You should be able to draw the graph through that point without lifting your pen.
How can a graph help visualize the Squeeze Theorem?
You can see the function being 'squeezed' between two other functions that converge to the same limit.
What does the graph of y = sin(x)/x tell us about its limit as x approaches 0?
The graph visually shows that as x approaches 0, the function approaches 1, even though the function is undefined at x = 0.
How does the slope of a tangent line on a graph relate to limits?
The slope of the tangent line at a point is the limit of the difference quotient as h approaches 0, which represents the derivative at that point.
How can you use a graph to determine infinite limits?
Look for vertical asymptotes. If the function approaches infinity (or negative infinity) as x approaches a certain value, then the limit is infinite.
How does a jump discontinuity affect the limit of a function?
The limit does not exist at a jump discontinuity because the left-hand limit and the right-hand limit are not equal.
