Relationship_Between_Limits_and_Continuity

1. Relationship Between Limits and
Continuity
Understanding the Connection in
Calculus

2. Introduction
• • Limits and continuity are foundational
concepts in calculus.
• • Limits describe approaching behavior of a
function.
• • Continuity ensures a function is smooth and
unbroken.

3. What is a Limit?
• • Describes the value a function approaches
as x gets close to a point.
• • Focuses on behavior near the point, not
necessarily at the point.
• • Example:
• lim(x → 2) (x² - 4)/(x-2) = 4

4. What is Continuity?
• • A function is continuous if its graph can be
drawn without lifting the pen.
• • Requirements for continuity at x = a:
• 1. f(a) is defined.
• 2. lim(x → a) f(x) exists.
• 3. lim(x → a) f(x) = f(a).

5. Relationship Between Limits and
Continuity
• • Limits are used to check for continuity.
• • A function is continuous at x = a if:
• lim(x → a) f(x) = f(a).
• • If the limit exists but is not equal to f(a), the
function is discontinuous.

6. Key Differences
• • Limit:
• - Describes approaching behavior.
• - May exist even if the function is undefined
at that point.
• • Continuity:
• - Requires the limit to exist and to equal the
function's value.
• - Ensures no gaps, jumps, or holes.

7. Example - Limit vs Continuity
• Function: f(x) = (x² - 4)/(x-2)
• • lim(x → 2) f(x) = 4
• • f(2) is undefined → Discontinuous at x = 2
• • Limit exists, but continuity fails due to
undefined point.

8. Summary
• • Limits describe where a function is heading
near a point.
• • Continuity ensures the function is unbroken
at that point.
• • Limits are necessary to check for continuity
but are not enough by themselves.
• • Continuity requires the limit to exist and
equal the function's value.

9. Questions?
• Feel free to ask any questions about limits and
continuity!