Estimating Limits from Graphs & Tables

Based on the table of values, the best estimate for lim(x→0) f(x) is 0. As x gets closer to 0 from both sides (values like -0.1, -0.01, -0.001 from the left and 0.001, 0.01, 0.1 from the right), f(x) gets closer to 0. The values approach 0 from above (0.01, 0.0001, 0.000001) on both sides, suggesting that 0 is the limiting value. Options A, C, and D are values that f(x) takes when x is farther from 0, not the limiting value.

We inspect the values around x=1. From left (0.9, 0.99), f approaches 2 (1.9 to 1.99). From right (1.001, 1.01), f approaches 2 (2.001 to 2.01). Both sides approach 2, so the limit is 2.

We evaluate the table for (e^t - 1)/t near t=0. Positive side (0.1 to 0.01) approaches 1 (1.10517 to 1.01005). Negative side (-0.1 to -0.01) approaches 1 (0.904837 to 0.99005). Convergence from both sides to 1 gives the limit as 1.

We consider how scale affects visibility. A wide x-range (-10 to 10) means each unit is small on the plot, so rapid changes or oscillations very close to x=0 may appear as a blur or solid line, missing the fact that the function doesn't settle to a value. Zooming in (narrower scale) reveals the true behavior for accurate limit estimation.

We use the values for ln(1+x) near x=0 (defined for x>-1). Positive side (0.1 to 0.01): 0.0953 to 0.00995, approaching 0. Negative side (-0.1 to -0.01): -0.1054 to -0.01005, approaching 0. Both sides to 0, limit is 0.

We compute the one-sided limits. The left-hand limit lim_{x→3⁻} f(x) uses f(x)=x for x≤3, so approaches 3. The right-hand limit lim_{x→3⁺} f(x) uses f(x)=6-x for x>3, so approaches 6-3=3. Since both agree on 3, the limit is 3.

We note that sin(1/x) is bounded by -1 ≤ sin(1/x) ≤1, so -|x| ≤ x sin(1/x) ≤ |x|. As x→0, both -|x| and |x| approach 0. By the squeeze theorem, since the function is squeezed between two functions approaching 0, the limit is 0.

We analyze the table for log10(x) near x=1. From left (0.9 to 0.99), values approach 0 from negative (-0.0458 to -0.00436). From right (1.01 to 1.1), approach 0 from positive (0.0043 to 0.0414, but closer values would be smaller). Both sides approach 0, so the limit is 0.

We review the table for sin(t)/t near t=0. Positive side (0.2 to 0.02) approaches 1 (0.9933 to 0.999933). Negative side (-0.2 to -0.02) approaches 1 (same by even-odd properties). Both sides to 1, limit is 1.

We start by identifying the need to express the limit analytically with the correct notation. The function is (x³ - 8)/(x-2), so the limit as x approaches 2 is written as lim_{x→2} of the expression. The arrow → indicates "approaches" rather than equals, and we do not evaluate at x=2 since the limit concerns values near 2, not at 2. Thus, the correct notation is lim_{x→2} (x³ - 8)/(x-2).

We assess the one-sided behaviors from the graph. The left-hand limit is 2, but the right-hand limit is 4. For the two-sided limit to exist, both must be equal. Since 2 ≠ 4, the limit does not exist.

We observe the graph of cos(x), which at x=0 is 1, but near 0, it approaches 1 from both sides slightly below for small positive and negative x, getting closer. Both sides agree on 1, so the limit is 1.

We observe the graph of 2^x, which at x=0 is 1, and near 0 approaches 1 from below on left, above on right, but both sides to 1. The limit is 1.

We analyze 1/(x-2)² near x=2. As x approaches 2 from left or right, (x-2)² is positive and small, so 1/(x-2)² becomes very large positive. Since it grows without bound (to +∞), the limit does not exist as a real number, due to unboundedness.

We begin by recalling the definition of a limit. The limit lim_{x→c} f(x) = R means that for any small positive number epsilon, there exists a delta such that if 0 < |x - c| < delta, then |f(x) - R| < epsilon. This means as x approaches c from values not equal to c, f(x) can be made as close to R as desired by choosing x sufficiently close to c. This interpretation does not require f(c) to be defined or equal to R, and it applies from both sides unless specified otherwise.