9. Practice Questions

1. Evaluate:

lim (𝑥² − 2𝑥 + 1)\n𝑥→3

To evaluate this limit, we start by substituting x with 3 in the expression 𝑥² - 2𝑥 + 1. This gives us:

(3)² - 2(3) + 1 = 9 - 6 + 1 = 4.

So, the limit is 4.

Imagine you're filling a glass with water. As you approach the brim with water, you can perfectly predict how much water will be in it as you reach the top. Here, substituting the exact value is like predicting the exact level of water when you fill it to the top.

2. Evaluate:

𝑥² − 16\nlim\n𝑥→4 𝑥 − 4

In this limit, we face an indeterminate form if we plug in 4 directly since both the numerator and denominator become zero. Therefore, we need to factor the numerator:

lim (𝑥 - 4)(𝑥 + 4)\n = 𝑥→4 (𝑥 - 4) By cancelling (𝑥-4) from the numerator and denominator, we simplify this to: l{im (𝑥 + 4)\n = 8} This is our limit since it's now defined at x = 4.

Think of trying to divide a pizza (numerator) with a certain number of people (denominator). If the number of people is the same as those who have a pizza slice, the division becomes tricky (0/0). But once you rearrange how many people get a slice, the division becomes clear—like simplifying the limit.

3. Determine if the limit exists:

|𝑥 − 2|\nlim\n𝑥→2 𝑥 − 2

In this case, we need to analyze what happens to the function as x approaches 2. The function becomes |2 - 2|, which equals 0. Thus, since approaching from both sides results in 0, this limit does exist and is equal to 0.

Imagine you're walking towards a door from both sides, and no matter whether you approach from the left or right, you reach the same spot exactly in front of the door—this indicates that the limit exists.

4. Estimate the limit from the table:

𝑥²−1\n𝑓(𝑥) = , as 𝑥 → 1\n𝑥−1

To estimate the limit using a table, we compute the values of f(x) as x approaches 1 from both sides. If we look around 1, we find:

At x = 0.9, f(x) ≈ 1 At x = 1.0, f(x) = 0 At x = 1.1, f(x) ≈ 1

This indicates that as we get closer to 1, the values converge to 1. We can thus estimate that lim_{x→1} f(x) = 1.

Imagine forecasting the temperature as the day progresses. As you check the readings at 3 PM and 4 PM, you notice they get closer to 25°C as you approach the peak hour (1 PM), showing how values close to a point can give you insights into the limit.

5. Graphically explain: Why does lim not exist?

𝑥→0 𝑥`

To determine if the limit exists, we must look at the graph of the function as it approaches 0. If, as x approaches 0, the function oscillates between two different values and the left-hand limit differs from the right-hand limit, this tells us the limit does not exist. Hence, lim_{x→0} does not converge to a single value.

Consider trying to listen to two different radio channels at once. If both frequencies are strong and overlapping, you can't clearly hear one single channel—the sound becomes a mess. This situation reflects why a limit can fail to exist.

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