Determine if a limit exists

Learning Objectives Define a limit and state the three conditions under which a limit fails to exist. Determine if a limit exists by calculating and comparing left-hand and right-hand limits. Identify when a limit fails to exist due to unbounded behavior, such as at a vertical asymptote. Identify when a limit fails to exist due to oscillating behavior. Analyze piecewise functions to determine if a limit exists at the points where the function's rule changes. Interpret a function's graph to determine the existence of a limit at a specific point. Distinguish between the value of a function at a point, f(c), and the limit of the function as x approaches c. Imagine two roads merging into one. If they don't meet at the exact same point and height, what happens to...

TermDefinitionExample LimitThe value that a function f(x) 'approaches' as the input x 'approaches' some value c. It describes the behavior of the function near a point, not necessarily at the point itself.For f(x) = x + 2, as x approaches 3, f(x) approaches 5. We write this as lim_{x→3} (x+2) = 5. Left-Hand LimitThe value a function approaches as x approaches a point 'c' from the left side (i.e., using values of x that are less than c). It is denoted as lim_{x→c⁻} f(x).For the function f(x) = |x|/x, the left-hand limit as x approaches 0 is lim_{x→0⁻} f(x) = -1, because for any x < 0, f(x) is -1. Right-Hand LimitThe value a function approaches as x approaches a point 'c' from the right side (i.e., using values of x that are greater than c). It is...

The Core Condition for Limit Existence The limit of f(x) as x approaches c is L, written as `lim_{x→c} f(x) = L`, if and only if `lim_{x→c⁻} f(x) = L` and `lim_{x→c⁺} f(x) = L`. This is the fundamental test for the existence of a limit. You must evaluate the limit from both the left and the right. If they are the same finite number, the limit exists and is equal to that number. Limit Non-existence: Disagreement If `lim_{x→c⁻} f(x) ≠ lim_{x→c⁺} f(x)`, then `lim_{x→c} f(x)` Does Not Exist (DNE). This happens at a 'jump' in the graph. If the function approaches two different values from the left and the right, no single limit exists. Limit Non-existence: Unbounded Behavior If `lim_{x→c⁻} f(x) = ±∞` or `lim_{x→c⁺} f(x) = ±∞`, then `lim_{x→c} f(x)` Does Not Exist...