Advanced One-Sided Limits with Rational, Trig, and Log Functions

For x approaching 2 from the right, we have x > 2. So x - 2 is positive. Thus, |x - 2| =x - 2. Substitute: (x - 2)/|x - 2| = (x - 2)/(x - 2) = 1 for all x > 2. As x approaches 2 from the right the expression stays 1, so the right-hand limit equals 1.

For x approaching 0 from the right, x is positive and very small. The reciprocal 1/x becomes a very large positive number. There is no finite limit; the expression grows without bound toward positive infinity. Therefore the right-hand limit is +∞.

For x ≠ 3, factor numerator: x² - 9 = (x - 3)(x + 3). For x near 3 but greater than 3, cancel (x - 3) to get x + 3. The expression equals x + 3 for x ≠ 3. Taking the right-hand limit as x→3⁺ gives 3 + 3 = 6. So the one-sided limit from the right is 6.

The function f(x) = sin(x)/x is an even function, meaning f(-x) = f(x). Since the standard limit as x→0 is 1, the limit as x approaches 0 from the left is also 1. Alternatively, using the small angle approximation for x near 0, sin(x) ≈ x, so the ratio approaches 1.

For x

Factor the denominator: x² - 16 = (x - 4)(x + 4). The expression becomes (x - 4)/[(x - 4)(x + 4)]. For x ≠ 4, we can cancel the (x - 4) terms, simplifying the function to 1/(x + 4). As x approaches 4 from the right, the denominator approaches 4 + 4 = 8. Thus, the limit is 1/8.

For x positive and approaching 0, ln(x) becomes large in magnitude and negative because the logarithm of a number less than 1 is negative and tends to negative infinity as the argument approaches 0+. Therefore the right-hand limit is -∞.

The graph shows constant value y = 2 for x values less than 0, so approaching 0 from the left the function values approach 2. For x > 0 the graph shows y = -1, so approaching 0 from the right the values approach -1. Therefore the left-hand limit is 2 and the right-hand limit is -1.

Factor numerator: x² - 4 = (x - 2)(x + 2). For x ≠ 2, the expression simplifies to x + 2. Approaching 2 from the left gives 2 + 2 = 4. The one-sided approach does not change the simplified value, so the left-hand limit equals 4.

Simplify the expression: (x - 1)/(x - 1)² = 1/(x - 1) for x ≠ 1. For x approaching 1 from the right, x - 1 is a small positive number, so 1/(x - 1) is a large positive number. Therefore the right-hand limit is +∞.

For x ≠ 0, √(x²) = |x|. So the expression equals x/|x|. For x

Factor numerator: x² - 5x + 6 = (x - 2)(x - 3). For x ≠ 2, the expression simplifies to x - 3. Substituting x = 2 into the simplified expression gives 2 - 3 = -1. Therefore, the limit is -1.

Factor the numerator: x² - 4 = (x - 2)(x + 2). For x ≠ 2, the expression simplifies to x + 2. Substituting x = 2 into the simplified expression gives 2 + 2 = 4. Therefore, the limit is 4.

As x approaches 0 from the left, 1 + x approaches 1. The reciprocal 1/(1 + x) therefore approaches 1. So the left-hand limit equals 1..

The definition of a limit existing at a point x = c requires that the limit approaching from the left and the limit approaching from the right both exist and are equal to the same value.