Limits, in a little moment

Silent study video. On-screen explanation:

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Getting closer is the point.
A limit describes what the output approaches as the input gets closer to a value. For f(x) = x², inputs near 2 give outputs near 4.
x → 2      ·      x² → 4
From the left: 1.9² = 3.61
Closer still: 1.99² = 3.9601
From the right: 2.01² = 4.0401
Source: Original calculus example notes · 1.3 Limits and continuity

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Two sides. One destination.
A finite two-sided limit exists when the left-hand and right-hand limits exist and agree. For x² near 2, both sides approach 4.
Left → 4       =       Right → 4
Same value: The two-sided limit exists.
Different values: The two-sided limit does not exist.
Source: Original calculus example notes · 1.3 Limits and continuity

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The limit can survive a hole.
This expression equals x + 2 when x ≠ 2. It is undefined at 2, yet its limit as x approaches 2 is still 4.
(x² − 4) / (x − 2) = x + 2,  x ≠ 2
At x = 2: The expression is undefined.
Near x = 2: The output approaches 4.
Source: Original calculus example notes · 1.3 Limits and continuity