Continuity and End Behavior

Develop continuity at a point, rational limits at infinity.

Lesson 2 of 32 parts Practice Flashcards

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Continuity at a Point

Objective: Explain continuity at a point, apply the rule, and justify relevant restrictions.

A function is continuous at a if f(a) is defined, the finite two-sided limit exists, and that limit equals f(a). A removable discontinuity can be repaired by assigning the limiting value at the point. A jump cannot be repaired with one value because its one-sided limits differ. These conditions separate missing values, nonexisting limits, and mismatched values.

Formulas

Continuity: limₓ→a f(x)=f(a)

Worked examples

Try itLet f(x)=(x²−49)/(x−7) for x≠7, and f(7)=k. Which k makes f continuous?Show answer

Step 1: Cancel away from x=7 to obtain x+7. Step 2: The limit at x=7 is 14. Step 3: Continuity requires k=14.

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Continuity at a Point

Objective: Explain continuity at a point, apply the rule, and justify relevant restrictions.

Formulas

Continuity: limₓ→a f(x)=f(a)

Worked examples

Try itLet f(x)=(x²−49)/(x−7) for x≠7, and f(7)=k. Which k makes f continuous?Show answer

Step 1: Cancel away from x=7 to obtain x+7. Step 2: The limit at x=7 is 14. Step 3: Continuity requires k=14.

Rational Limits at Infinity

Objective: Explain rational limits at infinity, apply the rule, and justify relevant restrictions.

For a rational function at large |x|, compare the highest powers of numerator and denominator. Equal degrees give the ratio of leading coefficients; a smaller numerator degree gives zero. Divide by the highest denominator power to justify the result. A horizontal asymptote describes end behavior and may be crossed at finite inputs; it is not a forbidden output value.

Formulas

$limₓ→∞(ax+b)/(cx+d)=a/c, c≠0$

Worked examples

Try itFind limₓ→+∞ (3x+7)/(5x+1).Show answer

Step 1: Divide numerator and denominator by x. Step 2: Obtain (3+7/x)/(5+1/x). Step 3: The terms involving 1/x tend to zero, leaving 3/5.

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