By plugging in values of x that get closer and closer to the target value and observing the trend of f(x).

By visually inspecting the graph and observing where the function is headed as x approaches the target value from both sides.

By using factoring, rationalizing, or trig identities to simplify the function and then using direct substitution.

When the left-hand limit and the right-hand limit are not equal, or when the function approaches infinity.

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.

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).

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.

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.

Estimating a limit involves approximation using tables or graphs, while finding the exact limit involves algebraic techniques.

To ensure that the limit exists, as both one-sided limits must be equal.

Limit Existing: Function approaches a specific value. | Function Continuous: Limit exists, function is defined, and limit equals the function value.

Removable: Limit exists, but function value is different or undefined. | Non-Removable: Limit does not exist (e.g., jump, asymptote).

Graphically: Visual approximation, can be less precise. | Numerically: Approximation using tables, precision depends on step size.

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.

Continuity: Function has no breaks or jumps. | Differentiability: Function has a derivative at every point (smooth curve).

Average: Slope of secant line over an interval. | Instantaneous: Slope of tangent line at a single point (derivative).

Direct Substitution: Plug in value directly; works if function is continuous. | Algebraic Manipulation: Simplify first (factoring, etc.) when direct substitution fails.

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).

At a Point: Examine function behavior near a specific x-value. | At Infinity: Examine function behavior as x becomes very large or very small.

IVT: Guarantees a specific function value within an interval. | MVT: Guarantees a specific derivative value within an interval.

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.

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.

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.

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.

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.

1. Set the two pieces of the function equal to each other at the point where they meet. 2. Solve for the unknown variable.

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.

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.

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.

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.