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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.
Explain how tables can be used to estimate limits.
By plugging in values of x that get closer and closer to the target value and observing the trend of f(x).
Explain how graphs can be used to estimate limits.
By visually inspecting the graph and observing where the function is headed as x approaches the target value from both sides.
Explain how algebraic manipulation can be used to find limits.
By using factoring, rationalizing, or trig identities to simplify the function and then using direct substitution.
When does a limit not exist?
When the left-hand limit and the right-hand limit are not equal, or when the function approaches infinity.
What is the relationship between limits and continuity?
For a function to be continuous at a point, the limit must exist at that point, the function must be defined at that point, and the limit must equal the function value.
Explain the concept of a one-sided limit.
The value a function approaches as the input approaches a certain value from either the left (left-hand limit) or the right (right-hand limit).
Explain the Squeeze Theorem.
If g(x) ≤ f(x) ≤ h(x) for all x near c, and lim x→c g(x) = lim x→c h(x) = L, then lim x→c f(x) = L.
Explain the Intermediate Value Theorem.
If f is a continuous function on the closed interval [a, b] and k is any number between f(a) and f(b), then there exists at least one number c in the interval (a, b) such that f(c) = k.
What is the difference between estimating a limit and finding the exact limit?
Estimating a limit involves approximation using tables or graphs, while finding the exact limit involves algebraic techniques.
Why is it important to check both the left and right-hand limits?
To ensure that the limit exists, as both one-sided limits must be equal.
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.