One-Sided & Infinite Limits in Context

The notation lim(x→∞) f(x) = L represents the limit of f(x) as x approaches infinity. This describes the behavior of the function as x becomes arbitrarily large (positive). The symbol "∞" indicates infinity, and the expression means that as x grows without bound, f(x) approaches the value L. Option B is incorrect because it describes x approaching zero, not infinity. Option C is incorrect because infinity is not a number that x can equal; it's a concept describing unbounded growth. Option D is incorrect because L is the value that f(x) approaches, not the value that x approaches.

The notation lim(x→a-) f(x) represents "the limit of f(x) as x approaches a from the left." The "-" symbol after "a" indicates that we're considering values of x that approach a from values less than a (from the left on the number line). Option B represents the limit from the right (values greater than a). Option C represents the limit as x approaches -a, not as x approaches a from the left. Option D uses words rather than standard mathematical notation.

If lim(x→3) f(x) = 4, it means that as x gets closer to 3, f(x) gets closer to 4. 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 = 3; f(3) could be 4, or it could be something else, or it could be undefined. Option C is incorrect because the function doesn't have to equal 4 for all x near 3; it just has to approach 4 as x approaches 3. Option D is incorrect because while the graph might have a hole at (3, 4), it could also pass through this point if f(3) = 4.

The notation lim(x→0+) 1/x = ∞ indicates that as x approaches 0 from the positive side, 1/x grows without bound. This means that as x gets closer and closer to 0 from values greater than 0, the values of 1/x become larger and larger without any upper limit. Option B is incorrect because infinity is not a number that the function can equal; it's a concept describing unbounded growth. Option C describes the behavior as x gets larger, not as x approaches 0. Option D is true about the function 1/x (it does have a horizontal asymptote at y = 0), but it doesn't describe what the given limit notation indicates.

If lim(x→2-) f(x) = 3 and lim(x→2+) f(x) = 3, then lim(x→2) f(x) = 3. For a two-sided limit to exist, the left-hand limit and right-hand limit must both exist and be equal. In this case, both one-sided limits exist and they are equal (both equal 3), so the two-sided limit exists and equals 3. Option A is incorrect because it uses the x-value instead of the y-value that the function approaches. Option C is incorrect because the limit does exist in this scenario. Option D is incorrect because we have enough information to determine the limit.

If the graph shows that as x approaches -1 from both sides, f(x) approaches 2, then lim(x→-1) f(x) = 2. The limit represents the y-value that the function approaches as x gets closer to -1. Since the function approaches the same value (2) from both the left and right sides, the two-sided limit exists and equals 2. Option A is incorrect because it uses the x-value instead of the y-value that the function approaches. Option B is incorrect because there's no information suggesting the function approaches 0. Option D is incorrect because the graph clearly shows the limiting behavior.

A graphical representation might fail to show that a limit does not exist because the scale might be too large to show oscillations or rapid changes. 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, suggesting a limit exists when it actually doesn't. Option A is incorrect because graphs don't always accurately represent limit behavior. Option C is incorrect because graphs can represent functions that are undefined at a point (often shown with a hole or asymptote). Option D is incorrect because even continuous functions can have points where limits don't exist if there are discontinuities.

A limit does not exist when the function approaches different values from the left and right. For a two-sided limit to exist, the function must approach the same value from both directions. If the left-hand limit and right-hand limit exist but are not equal, then the two-sided limit does not exist. Option A is incorrect because if a function is continuous at a point, the limit does exist there. Option B describes a condition where the limit does exist. Option D describes a situation where the limit exists.

If the graph shows that as x approaches 0 from both sides, f(x) approaches 0, then limx→0 f(x) = 0. The limit represents the y-value that the function approaches as x gets closer to 0. Since the function approaches the same value (0) from both the left and right sides, the two-sided limit exists and equals 0. Options A and C are incorrect because there's no information suggesting the function approaches -1 or 1. Option D is incorrect because the graph clearly shows the limiting behavior.

Based on the table of values, the best estimate for lim(x→1) f(x) is 3. As x gets closer to 1 from both sides (values like 0.9, 0.99, 0.999 from the left and 1.001, 1.01, 1.1 from the right), f(x) gets closer to 3. The values approach 3 from below (2.9, 2.99, 2.999) and from above (3.001, 3.01, 3.1), suggesting that 3 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 1, not the limiting value.

Based on the table, the best estimate for lim(x→0) g(x) is 0. As x gets closer to 0 from both sides, g(x) gets closer to 0. From the negative side (-0.1, -0.01, -0.001), g(x) approaches 0 from below (-0.1, -0.01, -0.001). From the positive side (0.001, 0.01, 0.1), g(x) approaches 0 from above (0.001, 0.01, 0.1). The pattern clearly shows that as x gets closer to 0, g(x) gets closer to 0. Options A and C are values 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, f(x) grows without bound" best describes the meaning of lim(x→c) f(x) = ∞. This captures the concept of an infinite limit - it describes the behavior of the function as x approaches c, where the function values become larger and larger without any upper bound. Option A is incorrect because infinity is not a value that the function can take at a point; it's a concept describing unbounded growth. Option C is incorrect because the function doesn't equal infinity for all x near c; it just grows without bound as x approaches c. Option D might be true (a vertical asymptote often corresponds to an infinite limit), but it's not the definition of what the limit notation means.

We begin by recalling the definition of a limit. The limit lim_{x→c} f(x) = L means that for any small positive number epsilon, there exists a delta such that if 0

A limit might fail to exist if 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 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.

If lim(x→3) f(x) = 4, then it must be true that as x gets closer to 3, f(x) gets closer to 4. 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 = 3; f(3) could be 4, or it could be something else, or it could be undefined. Option B is incorrect because for a function to be continuous at a point, the limit must equal the function value at that point, which we don't know from the given information. Option D is incorrect because the function doesn't have to equal 4 for all x near 3; it just has to approach 4 as x approaches 3.