DPST1013 Chap.6 Functions, Limits and Continuity
Functions, Limits and Continuity
Limits describe function behaviour near a point without requiring the value at that point. The epsilon-delta definition controls outputs through nearby inputs, while one-sided limits decide existence. Continuity adds the requirement that the function value exists and equals the limit, enabling the Intermediate Value and extrema theorems.
What this chapter covers
- 01
Function domains and combinations
- 02
Formal finite and infinite limits
- 03
Limit laws and indeterminate forms
- 04
One-sided limits
- 05
Continuity at a point
- 06
Intermediate Value and extrema theorems
Worked example · free
Choose a parameter for continuity
- 1Factor the left branch to x + 2 for nearby x not equal to 2.
- 1The left limit at 2 is 4.
- 1The right limit and function value are both 2k + 1.
- 1Solve 2k + 1 = 4 to obtain k = 3/2.
Key terms
- One-sided limit
- The limiting behaviour as inputs approach from only the left or only the right.
- Removable discontinuity
- A missing or mismatched function value where the finite two-sided limit still exists.
- Intermediate Value Theorem
- A continuity theorem guaranteeing every value between the endpoint outputs is attained.
- Pinching theorem
- A theorem that identifies a limit when a function is trapped between two functions with the same limit.
Functions, Limits and Continuity FAQ
Can a limit exist when the function is undefined?
Yes. A limit uses nearby values and excludes the point itself. A removable hole can therefore have a finite two-sided limit even though the function value is missing. Continuity fails until the value is defined appropriately.
What proves a two-sided limit exists?
The left-hand and right-hand limits must both exist and be equal. A plotted point or direct substitution alone cannot establish that condition, especially for piecewise functions and expressions with a zero denominator.
Does the Intermediate Value Theorem give a unique root?
No. Continuity and opposite endpoint signs guarantee at least one root between the endpoints. Uniqueness requires additional evidence, such as strict monotonicity, because a continuous graph may cross the axis several times.
What should I do after direct substitution gives zero over zero?
Treat the result as an indeterminate form, not as the value zero and not as proof that the limit fails. Factor, rationalise, combine fractions or use a standard limit to reveal the local behaviour. Cancel only a factor that is non-zero near the point, while remembering that the original expression may remain undefined at the point itself. Then evaluate the simplified nearby expression.
How are infinite limits different from limits at infinity?
An infinite limit describes unbounded output as the input approaches a finite location, often signalling a vertical asymptote. A limit at infinity describes long-run behaviour as the input grows without bound, often identifying a horizontal or oblique asymptote. The phrases concern different variables and different graph features. Write the approach notation explicitly before applying algebraic techniques.
When may limit laws be applied to a quotient?
The numerator and denominator limits must exist in the relevant sense, and the denominator limit must be non-zero for the ordinary quotient law. If the denominator tends to zero, the law does not decide the result. Examine signs, one-sided behaviour or algebraic cancellation instead. Stating the non-zero condition prevents a common misuse that turns an indeterminate expression into an unjustified fraction.
How do I make a piecewise function continuous at a joining point?
Calculate the left-hand limit from the branch used on the left and the right-hand limit from the branch used on the right. Set them equal to determine any unknown parameter. Then ensure the defined function value at the join equals that common limit. Matching only the two formulas at an arbitrary input or checking just one side does not establish continuity.
What does the squeeze theorem require?
Find lower and upper functions that bound the target near the point, and show that both bounding functions have the same limit. The target then shares that limit. The inequalities need only hold sufficiently close to the point, not necessarily at it. This method is especially useful when a bounded oscillating factor is multiplied by something whose magnitude tends to zero.
Exam move
Separate the function value from nearby behaviour on every limit problem. Factor or rationalise before declaring a zero-over-zero form unresolved. For piecewise functions, compute both one-sided limits and then compare with the actual value. When using a theorem, state its interval and hypotheses before the conclusion.
Build simple counterexamples, such as a removable hole or step function, to test logical implications between limits, values and continuity. Sort practice by the obstruction revealed after substitution: removable factor, radical cancellation, unequal one-sided behaviour, infinite growth or oscillation. This makes technique selection evidence-based.
Annotate graphs with open points, filled points and approach arrows, but confirm every graphical claim algebraically when a formula is supplied. For each theorem question, underline the exact conclusion requested and list every hypothesis separately. Revisit errors by constructing a nearby example where the tempting shortcut produces the wrong result.
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