Find one-sided limits using graphs

Learning Objectives Define a one-sided limit from the left and from the right. Interpret the mathematical notation for left-hand (x→c⁻) and right-hand (x→c⁺) limits. Visually determine the value of a left-hand limit by tracing a function's graph. Visually determine the value of a right-hand limit by tracing a function's graph. Identify from a graph when a one-sided limit does not exist due to unbounded behavior (approaching ±∞). Distinguish between the value of a one-sided limit and the value of the function at a point, f(c). Compare left-hand and right-hand limits to draw conclusions about the existence of the overall two-sided limit. Imagine you're hiking on a path represented by a graph. What altitude are you approaching as you get infinitely close to a c...

TermDefinitionExample One-Sided LimitThe y-value that a function approaches as the x-value gets closer to a specific point from only one side (either from the left or from the right).The speed a car approaches as it nears a stop sign, considered only from the moments before it stops. Left-Hand LimitThe y-value a function f(x) approaches as x gets closer to a number 'c' from values less than 'c'.Notation: lim_{x→c⁻} f(x). To find lim_{x→2⁻} f(x), we look at x-values like 1.9, 1.99, 1.999, etc. Right-Hand LimitThe y-value a function f(x) approaches as x gets closer to a number 'c' from values greater than 'c'.Notation: lim_{x→c⁺} f(x). To find lim_{x→2⁺} f(x), we look at x-values like 2.1, 2.01, 2.001, etc. Jump DiscontinuityA point on a graph where t...

Left-Hand Limit Notation lim_{x→c⁻} f(x) = L This reads 'the limit of f(x) as x approaches c from the left is L'. The superscript minus sign (-) indicates approach from the left side (values smaller than c). Right-Hand Limit Notation lim_{x→c⁺} f(x) = R This reads 'the limit of f(x) as x approaches c from the right is R'. The superscript plus sign (+) indicates approach from the right side (values larger than c). Existence of a Two-Sided Limit lim_{x→c} f(x) = L ⟺ lim_{x→c⁻} f(x) = L and lim_{x→c⁺} f(x) = L The general (two-sided) limit exists and equals L if and only if the left-hand limit and the right-hand limit both exist and are equal to L.