Evaluate limits and test continuity

A function is undefined at x = 1. What can you conclude about its limit there?

Original function

\(\displaystyle f(x)=\frac{x^2-1}{x-1},\quad x\ne1\)

Reduce h through all ten settings. The original quotient is undefined at input 1; compare values on either side

What do both columns of function values approach?

Substitution or simplification

Direct substitution works for a polynomial. A quotient giving zero over zero needs algebra first; zero over zero is not a numerical answer. A limit asks about nearby inputs, so cancellation away from the excluded target can establish it

1. 1Nearby behaviour, not automatically the point value A limit describes what function values approach as the input approaches a target. The function may be undefined at the target or have a different value there
2. 2Simplify before substituting into an indeterminate quotient Factor the complete numerator and denominator. Cancel a common factor only away from its zero, then evaluate the simpler nearby rule \(\displaystyle \frac{(x-a)Q(x)}{x-a}=Q(x),\quad x\ne a\)

Given

\(\displaystyle \lim_{x\to-1}(2x^2-x+3)\)

Evaluate this polynomial limit

Given

\(\displaystyle \lim_{x\to2}\frac{x^2-4}{x-2}\)

Factor, cancel away from the excluded input and evaluate the limit

What is the original function value at input 2?

Read the branch condition first

For x < a use the left branch; for x > a use the right branch. An equality branch, when present, gives only the point value. Strict inequalities leave the join out of those nearby branches

1. 1Compare both sides A finite two-sided limit exists when the left and right limiting values are equal \(\displaystyle \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\)

Given

\(\displaystyle f(x)=2x+1\text{ for }x<1,\quad\lim_{x\to1^-}f(x)\)

Find the left-hand limit from the stated branch rule

Given

\(\displaystyle f(x)=x+4\text{ for }x>1,\quad\lim_{x\to1^+}f(x)\)

Find the right-hand limit from the stated branch rule

The left limit is 3, the right limit is 5 and f(1) = 7. What is the two-sided limit?

Test continuity only after comparing the limits

A hole has no point value. A redefined point may have a value different from its nearby limit. A jump has unequal one-sided limits. None is continuous at the join

1. 1Continuity has three conditions The function value must exist, the finite two-sided limit must exist, and they must agree \(\displaystyle f(a)=\lim_{x\to a}f(x)\)

Both one-sided limits equal −2, but f(0) = 3. Which conclusion follows?

A rational limit at infinity

An unbounded input is not a number to substitute. Divide by the highest denominator power and compare terms that survive

1. 1At an unbounded input, compare dominant powers For the rational cases here, divide by the highest denominator power. Lower-degree numerator terms vanish; equal degrees give the ratio of leading coefficients

Given

\(\displaystyle \lim_{x\to-\infty}\frac{2x+1}{x^2+3}\)

Compare the degrees and evaluate this limit

Signed unbounded limits

For the reciprocal square function, both sides grow without bound positively as x approaches zero. For the reciprocal function, the left side tends to −∞ and the right to +∞. The questions state the extended-real reporting convention

1. 1Report unbounded behaviour as a classification Under the stated convention, matching signed unbounded one-sided behaviours give +∞ or −∞. Opposite signs give no two-sided limit. Infinity is never entered as a finite number

Under the stated extended-real convention, what is the two-sided limit as x approaches zero?