Both one-sided limits at 6 equal -4 and f(6) = 9. Which listed change preserves the two-sided limit -4?

• Change only f(6) from 9 to 0.
• Change the left-hand limit to 0, keeping the right-hand limit at -4.
• Change the right-hand limit to 0, keeping the left-hand limit at -4.
• Change both one-sided limits from -4 to the new limiting value 0.

What is the limit of |x|/x as x approaches 0 from the right?

• -1
• 0
• 1

If both one-sided limits exist as finite numbers and agree, the finite two-sided limit equals their shared value.

What is the limit of |x|/x as x approaches 0 from the left?

• -1
• 0
• 1

Which supplied function has unequal finite one-sided limits at 7?

• f(x) = x+2 on both sides
• f(x) = 2 below 7 and 5 above 7
• f(x) = x² on both sides
• f(x) = 4 on both sides

If the left side keeps oscillating without approaching any value or infinite direction, a right-hand limit of 3 does not establish a two-sided limit.

The left limit is 8 and the right limit is 2. Someone reports a limit of 5 by averaging. Why does that calculation not establish a limit?

• Averaging always gives the larger approaching value.
• A two-sided limit uses only the right-hand approaching value.
• The assigned point value must equal the arithmetic mean.
• Averaging does not make the approaching values agree.

At 3 the left limit is -2 and the right limit is 7. Which assignment of f(3) makes the two-sided limit exist?

• Set f(3) equal to the left limit -2.
• Set f(3) equal to the right limit 7.
• Set f(3) equal to their mean 2.5.
• No assignment changes the unequal limits.

For f(x) = 1/x at 0, why is there no two-sided infinite limit?

• Both sides approach zero as x approaches zero.
• The sides tend to opposite infinities.
• The sides both approach positive infinity.

For x distinct from 0, let h(x) = 1/x + 1/|x|. What happens as x approaches 0?

• Both sides approach positive infinity.
• Both sides approach the finite value zero.
• The left approaches zero and the right positive infinity.
• The left approaches negative infinity and the right zero.