Assume domain inputs other than a occur arbitrarily close to a. Call x permitted when it is in the domain and 0 < |x-a| < delta. The required bound is |f(x)-L| < epsilon. Which statement defines limit L?

• For every epsilon > 0, some delta > 0 makes the bound hold for every permitted x.
• For every delta > 0, some epsilon > 0 makes the bound hold for every permitted x.
• Some epsilon > 0 has a delta > 0 making the bound hold for every permitted x.

For f(x) = 5x+1 near a = 2, the target is L = 11. With delta = 0.03, what is the boundary output-error bound 5 delta? Give a decimal.

Epsilon is fixed. A proposed proof chooses a new delta after seeing each input x. Which requirement has not been met?

• The proof must set delta equal to epsilon.
• The proof must include the input x = a.
• One delta must work for all permitted inputs.
• One input must have output exactly equal to L.

With a and L fixed, which quantity is chosen first in the epsilon-delta test?

• The distance delta around a
• The tolerance epsilon around L
• The value f(a) at the point

For f(x) = -2x+4 near a = 1 with target L = 2, epsilon = 0.12 is given. Which comparison verifies the choice delta = 0.03?

• 2 times 0.03 = 0.06 < 0.12
• 2 times 0.03 = 0.12 < 0.06
• -2 times 0.03 = 0.06 > 0.12
• 0.03 divided by 2 = 0.12

For the same function, point and target, a delta that works for epsilon = 0.002 also works for epsilon = 0.02.

Let f(x) = 8 for every real x. For its limit 8 at a = 3 and epsilon = 0.1, which statement about delta is correct?

• Only delta equal to 0.1 works.
• Only delta less than 0.1 works.
• No positive value of delta works.
• Every positive delta works.

Finding a working delta for one small epsilon proves a limit.

To prove 3x + 1 tends to 7 as x tends to 2 for an arbitrary epsilon, which delta completes the proof?

• 3 times epsilon
• epsilon / 3
• epsilon + 3

Let f(x) = x squared, with limit 0 as x tends to 0. For epsilon = 0.25, give the largest delta that works.