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How to find $\lim_{x \to a} f(x)$ given a graph?
1. Locate 'a' on the x-axis. 2. Trace the graph from the left towards x = a. Note the y-value. 3. Trace the graph from the right towards x = a. Note the y-value. 4. If both y-values are the same, that's the limit. If not, the limit DNE.
How to find $\lim_{x \to a} f(x)$ given a table of values?
1. Look at x-values approaching 'a' from the left. Note the trend in f(x). 2. Look at x-values approaching 'a' from the right. Note the trend in f(x). 3. If f(x) approaches the same value from both sides, that's the limit.
How to find $\lim_{x \to a} f(x)$ algebraically when direct substitution yields an indeterminate form?
1. Try factoring, rationalizing, or using trig identities to simplify the expression. 2. Cancel out any common factors. 3. Use direct substitution on the simplified expression.
How to determine if a piecewise function is continuous at a point?
1. Check if the left-hand limit and the right-hand limit are equal at that point. 2. Check if the function is defined at that point. 3. Check if the limit equals the function value at that point.
How to evaluate $\lim_{x \to \infty} \frac{P(x)}{Q(x)}$ where P and Q are polynomials?
1. Divide both the numerator and denominator by the highest power of x in the denominator. 2. Evaluate the limit as x approaches infinity. Terms of the form c/x^n approach 0.
How to find the value that makes a piecewise function continuous?
1. Set the two pieces of the function equal to each other at the point where they meet. 2. Solve for the unknown variable.
How do you use the Squeeze Theorem to find a limit?
1. Find two functions, g(x) and h(x), such that g(x) ≤ f(x) ≤ h(x). 2. Find the limits of g(x) and h(x) as x approaches c. 3. If both limits are equal to L, then the limit of f(x) as x approaches c is also L.
How to determine if a function is differentiable at a point?
1. Check if the function is continuous at the point. 2. Find the left and right derivatives at the point. 3. If the left and right derivatives are equal, the function is differentiable at the point.
How to handle limits involving absolute values?
1. Rewrite the absolute value function as a piecewise function. 2. Evaluate the left and right-hand limits separately. 3. If the left and right-hand limits are equal, the limit exists and is equal to that value.
How to find limits of trigonometric functions?
1. Try direct substitution. 2. If direct substitution results in an indeterminate form, use trigonometric identities to simplify the expression. 3. Apply special trigonometric limits, such as lim x→0 (sin x)/x = 1.
What are the differences between a limit existing and a function being continuous at a point?
Limit Existing: Function approaches a specific value. | Function Continuous: Limit exists, function is defined, and limit equals the function value.
What are the differences between removable and non-removable discontinuities?
Removable: Limit exists, but function value is different or undefined. | Non-Removable: Limit does not exist (e.g., jump, asymptote).
What are the differences between estimating limits graphically vs. numerically?
Graphically: Visual approximation, can be less precise. | Numerically: Approximation using tables, precision depends on step size.
What are the differences between one-sided limits and two-sided limits?
One-sided: Limit from the left or right only. | Two-sided: Limit from both left and right must be equal for the limit to exist.
What are the differences between continuity and differentiability?
Continuity: Function has no breaks or jumps. | Differentiability: Function has a derivative at every point (smooth curve).
What are the differences between average rate of change and instantaneous rate of change?
Average: Slope of secant line over an interval. | Instantaneous: Slope of tangent line at a single point (derivative).
What are the differences between direct substitution and algebraic manipulation for finding limits?
Direct Substitution: Plug in value directly; works if function is continuous. | Algebraic Manipulation: Simplify first (factoring, etc.) when direct substitution fails.
What are the differences between a secant line and a tangent line?
Secant Line: A line that intersects a curve at two points. | Tangent Line: A line that touches a curve at only one point (instantaneous rate of change).
What are the differences between finding limits at a point versus at infinity?
At a Point: Examine function behavior near a specific x-value. | At Infinity: Examine function behavior as x becomes very large or very small.
What are the differences between the Intermediate Value Theorem and the Mean Value Theorem?
IVT: Guarantees a specific function value within an interval. | MVT: Guarantees a specific derivative value within an interval.
What is a limit?
The value that a function approaches as the input approaches some value.
What is a removable discontinuity?
A point on a graph that is not defined, but the limit exists at that point.
What does it mean for a function to be continuous at a point?
The limit exists at the point, the function is defined at the point, and the limit equals the function value at the point.
Define left-hand limit.
The value a function approaches as the input approaches a certain value from the left side.
Define right-hand limit.
The value a function approaches as the input approaches a certain value from the right side.
What is the formal definition of a limit?
For every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε.
What is an indeterminate form?
An expression whose limit cannot be evaluated by direct substitution, such as 0/0 or ∞/∞.
What is direct substitution?
Evaluating a limit by plugging in the value that x approaches into the function.
What is a piecewise function?
A function defined by multiple sub-functions, each applying to a certain interval of the domain.
What is a non-removable discontinuity?
A discontinuity where the limit does not exist, such as a jump or vertical asymptote.