How do you determine the left-hand limit of a function from its graph when there is a vertical asymptote?

To find the left-hand limit as x approaches a value c (denoted

\lim_{x \to c^-} f(x)

), observe the behavior of the function's curve strictly for input values less than c, moving towards c. If the y-values grow without bound in either the positive or negative direction, the finite real limit does not exist. In such cases, one may describe the specific unbounded behavior using infinite-limit notation (e.g.,

+\infty

-\infty

Conditions

The function has a vertical asymptote or discontinuity at $x=c$ .
You are evaluating the limit from the left side (inputs strictly less than c).

Reasoning, step by step

Identify the target x-value on the horizontal axis.
Trace the graph along the branch where inputs are smaller than the target.
Observe whether the output y-values settle near a specific finite number or diverge.
If they diverge upwards indefinitely, conclude that no finite real limit exists; optionally write the result as $+\infty$ to specify upward divergence.

Example

In the video, finding $\lim_{x \to 6^-} g(x)$ involves tracing the blue curve from $x=5.75$ towards $x=6$ . The y-values shoot up past 9 and continue rising alongside the dashed vertical line at $x=6$ , indicating unbounded growth.

Common misconceptions

Believing that an infinite limit means the limit equals a specific large number like 1000.
Confusing the left-hand limit with the actual function value at $x=c$ (which might be undefined).