A limit describes the value that a function approaches as the input approaches some value. It focuses on the behavior near a point, not necessarily at the point itself.

One-sided limits examine the behavior of a function as it approaches a value from either the left (left-hand limit) or the right (right-hand limit).

A limit does not exist if the function is unbounded, oscillates wildly, or if the left-hand limit and right-hand limit are not equal.

By observing the y-value that the function approaches as x approaches a specific value. Trace the curve from both sides to see if they converge to the same y-value.

A jump discontinuity indicates that the left-hand limit and the right-hand limit are different, therefore the limit at that point does not exist.

A vertical asymptote indicates that the function is unbounded as x approaches a certain value, meaning the limit does not exist at that point.

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 scale of a graph can distort the appearance of the function, potentially hiding important details or exaggerating certain behaviors, leading to incorrect limit estimations.

Checking one-sided limits is crucial because the overall limit exists only if both the left-hand limit and the right-hand limit exist and are equal.

Oscillation refers to a function fluctuating rapidly between two values as x approaches a specific point. If the oscillations become infinitely rapid, the limit does not exist.

The value a function approaches as the input approaches a specific point.

The value a function approaches as x approaches a value from the left.

The value a function approaches as x approaches a value from the right.

Does Not Exist. The limit does not approach a single, finite value.

A vertical line that a function approaches but never touches; often indicates a limit DNE.

A point where the left-hand limit and right-hand limit exist but are not equal.

A function that fluctuates rapidly between two values as x approaches a certain point.

A function whose values increase or decrease without limit as x approaches a certain point.

$$\lim_{x \to a} f(x)$$

The limit exists at that point, the function is defined at that point, and the limit equals the function value.

1. Visualize the point. 2. Trace along the graph from both sides. 3. Check if the one-sided limits match.

1. Check for vertical asymptotes. 2. Check for jump discontinuities. 3. Check for wild oscillations.

Trace the graph from the left side of x=a. Determine the y-value the function approaches as x gets closer to a from the left.

Trace the graph from the right side of x=a. Determine the y-value the function approaches as x gets closer to a from the right.

Determine $$\lim_{x \to a^-} f(x)$$ and $$\lim_{x \to a^+} f(x)$$. If they are equal, that is the limit. Otherwise, the limit DNE.

The limit can still exist even with a hole. Focus on what y-value the function approaches as x approaches 'a' from both sides, not the value at x=a.

Check if the limit exists at x=a, if f(a) is defined, and if $$\lim_{x \to a} f(x) = f(a)$$.

Look for a point where the graph 'jumps' from one y-value to another. The left and right limits will be different at this point.

Look for a vertical line where the function approaches infinity (or negative infinity) as x approaches that line.

If the function oscillates wildly near a point, the limit likely does not exist. The function does not approach a single, finite value.