Understanding Limit Notation & Meaning

The correct notation for "the limit of f(x) as x approaches 2 is 5" is lim(x→2) f(x) = 5. In this notation, "lim" indicates we're taking a limit, "x→2" shows that x is approaching 2, and "= 5" indicates that the function value approaches 5. Option B incorrectly states that the function value at x=2 is 5, which is different from the limit. Option C swaps the values, showing x approaching 5 instead of 2. Option D uses arrow notation but incorrectly swaps the values.

If lim(x→0) f(x) = 1, it means that as x gets closer to 0, f(x) gets closer to 1. This is the fundamental meaning of a limit - it describes the behavior of the function as x approaches a particular value. Option A is incorrect because the limit doesn't tell us the value of the function at x = 0; f(0) could be 1, or it could be something else, or it could be undefined. Option C is incorrect because the function doesn't have to equal 1 for all x near 0; it just has to approach 1 as x approaches 0. Option D is incorrect because while the graph might have a hole at (0, 1), it could also pass through this point if f(0) = 1.

The notation lim(x→4) f(x) = 7 means that as x gets closer and closer to 4, f(x) gets closer and closer to 7. This is the fundamental concept of a limit - it describes the behavior of a function as the input approaches a particular value, not necessarily the value of the function at that point. Option A is incorrect because it confuses the limit with the function value. Option C swaps the roles of x and f(x). Option D suggests approximation rather than the precise limiting behavior.

The limit statement lim(x→0) sin(x)/x = 1 tells us that as x gets closer to 0, the value of sin(x)/x gets closer to 1. This describes the behavior of the function as x approaches 0, regardless of what happens exactly at x = 0. Option A is incorrect because sin(x)/x is actually undefined at x = 0 (division by zero). Option C is a true statement, but it only describes the function's status at the point x=0. The limit statement describes the function's behavior as x approaches 0, not just at the point itself. Option D is incorrect because it describes behavior as x gets larger, not as x approaches 0.

The two-sided limit exists if and only if both one-sided limits exist and are equal. This is a fundamental property of limits. For lim(x→c) f(x) to exist, both lim(x→c-) f(x) and lim(x→c+) f(x) must exist and have the same value. Option B is incorrect because having just one one-sided limit is not sufficient for the two-sided limit to exist. Option C is incorrect because the two-sided limit isn't an average but must equal both one-sided limits. Option D is incorrect because there is a direct relationship between one-sided and two-sided limits.

If the graph of f(x) approaches y = 3 as x approaches 2 from both sides, then lim(x→2) f(x) = 3. The limit represents the y-value that the function approaches as x gets closer to 2. Since the function approaches the same value (3) from both the left and right sides, the two-sided limit exists and equals 3. Option A (0) and Option B (2) are incorrect because they don't match the y-value that the function approaches. Option D is incorrect because the graph clearly shows the limiting behavior.

If the graph shows that as x approaches 1 from both the left and right sides, f(x) approaches 4, then lim(x→1) f(x) = 4. For a two-sided limit to exist, the function must approach the same value from both directions. In this case, since f(x) approaches 4 from both sides, the limit exists and equals 4. Option A is incorrect because it uses the x-value instead of the y-value that the function approaches. Options C and D are incorrect because the limit does exist in this scenario.

A graphical representation might miss important behavior because the scale of the graph could be too large to show rapid changes or oscillations near a point. For example, if a function oscillates rapidly between -1 and 1 as x approaches 0, but the graph is plotted with a scale where each unit on the x-axis represents a large interval, these oscillations might not be visible and could appear as a single line. Option A is too vague and doesn't specifically address why important behavior might be missed. Option B is incorrect because graphs can represent vertical asymptotes. Option D is incorrect because graphs are not always accurate representations, especially when important behavior occurs at a scale that isn't visible.

A limit does not exist when the function oscillates infinitely as x approaches the point. This is one of the ways a limit can fail to exist - if the function doesn't approach a single value but instead oscillates between multiple values infinitely often as x gets closer to the point. Option A is incorrect because a function can be defined at a point and still have a limit there (in fact, if a function is continuous at a point, it is defined there and the limit exists). Option B describes a condition where the limit does exist. Option D also describes a situation where the limit exists.

If lim(x→3-) f(x) = 5 and lim(x→3+) f(x) = 7, then lim(x→3) f(x) does not exist. For a two-sided limit to exist, the left-hand limit and right-hand limit must both exist and be equal. In this case, although both one-sided limits exist, they are not equal (5 ≠ 7), so the two-sided limit does not exist. Options A and B are incorrect because they only consider one side of the limit. Option C is incorrect because the limit isn't an average of the one-sided limits; it must equal both of them to exist.

Based on the table of values, the best estimate for lim(x→2) f(x) is 5. As x gets closer to 2 from both sides (values like 1.9, 1.99, 1.999 from the left and 2.001, 2.01, 2.1 from the right), f(x) gets closer to 5. The values approach 5 from below (4.9, 4.99, 4.999) and from above (5.001, 5.01, 5.1), suggesting that 5 is the limiting value. Option A is the x-value, not the y-value that f(x) approaches. Options B and D are values that f(x) takes when x is farther from 2, not the limiting value.

Based on the table, the best estimate for lim(x→0) g(x) is 1. As x gets closer to 0 from both sides, g(x) gets closer to 1. From the negative side (-0.1, -0.01, -0.001), g(x) approaches 1 from below (0.99, 0.9999, 0.999999). From the positive side (0.001, 0.01, 0.1), g(x) also approaches 1 from below (0.999999, 0.9999, 0.99). The pattern clearly shows that as x gets closer to 0, g(x) gets closer to 1. Option A is incorrect because g(x) is not approaching 0. Option B is a value that g(x) takes when x is farther from 0, not the limiting value. Option D is incorrect because the table provides enough information to estimate the limit.

The statement "As x gets arbitrarily close to c (but not equal to c), f(x) gets arbitrarily close to L" best describes the meaning of lim(x→c) f(x) = L. This captures the essential concept of a limit - it describes the behavior of the function as x approaches c, not necessarily the value at c itself. The limit is concerned with values of x arbitrarily close to c but not equal to c. Option A is incorrect because the limit doesn't depend on the function value at c. Option C is incorrect because the function doesn't have to equal L for all x near c, just approach L as x approaches c. Option D is incorrect because the graph may not pass through (c, L), especially if f(c) is undefined or different from L.

A limit does not exist when the function is unbounded as x approaches the point. This is one of the ways a limit can fail to exist - if the function grows without bound (positively or negatively) as x gets closer to the point, rather than approaching a specific finite value. Option A describes a situation where the limit does exist. Option B is not a reason for a limit to fail to exist; a function can be defined at a point and still have a limit there. Option D describes a situation where the limit exists.

The function being undefined at the point is NOT a way that a limit might fail to exist. A limit describes the behavior of a function as x approaches a point, not the value at the point itself. A function can be undefined at a point and still have a limit there. For example, lim(x→0) (x²)/x = 0, even though the function is undefined at x = 0. Options A, B, and C are all ways that a limit might fail to exist: if the function is unbounded, if it oscillates infinitely, or if the left and right limits are not equal.