Estimating Limits from Tables

In Calculus, limits describe the value a function approaches as the input gets close to a specific point. When algebraic evaluation is difficult or results in indeterminate forms, limits can be estimated using tables.

For a function f(x), the limit at x = c is written as: \lim_{x \to c}f(x)

Steps for calculating limits through tables: 

1. Start from the points which are infinitely close to the target point.
2. Approach the target point from left and go infinitely close while evaluating the function for every point.
3. Repeat the same thing from right-hand side.

The table that is created will have values that are almost equal, and will give an estimate of the limit at that point. 

Key Observations

• If values from both sides approach the same number L, then:\lim_{x \to c} f(x) = L
• If the values approach different numbers, the limit does not exist
• If values increase or decrease without bound, the limit is infinite (unbounded)
• The limit depends on the approaching values, not necessarily the value of the function at that point

Example: f(x) = x - 1, calculating the limit for this function at x = 0. 

f(x)=\lim_{x \to 0}x - 1

Let's calculate this value using tables. The way to calculate the value of the function at different values of x. The goal is to reach infinitely close to the target point but not at the target point. Keep in mind that while creating the table, the function should be approached from both the left-hand side and right-hand side.

x	f(x)
-0.1	-1.1
-0.05	-1.05
-0.001	-1.001
0.001	-0.999
0.05	0.95

In the table, as we move closer and closer to x = 0 from either side, the value of the function approaches the value -1. 

Thus,  \lim_{x \to 0}x - 1 = -1

While populating the table, some things must be kept in mind to get the right value of limit: 

1. Do not assume that the function value is the value of the limit. Sometimes there might be a discontinuity in the function, it might seem that function is going to take a particular value, but the actual value at that point is different. It often happens at the points where the function is either discontinuous or undefined.
2. Approach from both sides of the point.
3. Always go as close as possible to the point.

One-sided Limit from Tables

If only one-sided behavior is required:

• For x \to c^+ : take values greater than c
• For x \to c^- : take values less than c

Example: Consider an example, for the function f(x) = x². Calculate \lim_{x \to 0^{+}}f(x)

\lim_{x \to 0^{+}}f(x)

This means that value of right-hand side limit is asked. In this case, the table is populated only from the values which lie on the right-hand side of x = 0. 

x	f(x)
0.1	0.01
0.05	0.0025
0.001	0.000001
0.0005	0.00000025
0.0001	0.00000001

Notice from the table that the value of the limit is approaching the value 0. \lim_{x \to 0^{+}}f(x) = 0

Solved Problems

Question 1: Consider an example, for the function f(x) = x² .Calculate \lim_{x \to 1^+}f(x)

\lim_{x \to 1^+}f(x)

This means that value of right-hand side limit is asked. In this case, the table is populated only from the values which lie on the right-hand side of x = 1. 

x	f(x)
1.1	1.21
1.01	1.0201
1.001	1.002001
1.0001	1.00020001

Notice from the table that the value of the limit is approaching the value 1. \lim_{x \to 1^+}x^2 = 1

Question 2: Consider an example, for the function f(x) = 5x. Calculate \lim_{x \to 0}f(x)

\lim_{x \to 0}f(x)

In this case, the table is populated only from the values which lie on the left and right-hand side of x = 0. 

x	f(x)
-0.1	-0.5
-0.05	-0.25
-0.001	-0.005
0.001	0.005
0.01	0.05

Notice from the table that the value of the limit is approaching the value 0. \lim_{x \to 0}5x = 0

Question 3: Consider an example, for the function f(x) = \frac{x^2 - 1}{x - 1}  . Calculate \lim_{x \to 1^+}f(x)

\lim_{x \to 1^+}f(x)

This means that value of right-hand side limit is asked. In this case, the table is populated only from the values which lie on the right-hand side of x = 1. 

x	f(x)
0.999	1.999
0.9999	1.9999
0.99999	1.99999
1.00001	2.00001
1.0001	2.0001

Notice from the table that the value of the limit is approaching the value 2. \lim_{x \to 1^+}\frac{x^2-1}{x-1}= 2

Question 4: Consider an example, for the function f(x) = \frac{sin(x)}{x}  . Calculate \lim_{x \to 0}f(x)

\lim_{x \to 0}f(x)

In this case, the table is populated from the values which lie on the left and right-hand side of x = 0. 

x	f(x)
-0.1	0.99833
-0.01	0.99998
-0.001	0.99999
0.001	0.99999
0.01	0.99998

Notice from the table that the value of the limit is approaching the value 1. \lim_{x \to 0}f(x) = 1

Question 5: Consider an example, for the function f(x) = log(x). Calculate \lim_{x \to 1}f(x)

\lim_{x \to 1}f(x)

In this case, the table is populated from the values which lie on the left and right-hand side of x = 1. 

x	f(x)
0.999	0.000434
0.9999	0.0000434
0.99999	0.00000434
1.00001	0.00000434
1.0001	0.0000434

Notice from the table that the value of the limit is approaching the value 0. \lim_{x \to 1}log(x) = 0

Practice Problems

1. Find the value of the limit of the function f(x) = \lim_{x \to 3} (2x- 3)

2. Find the value of the limit of the function f(x) = \lim_{x \to -2} (x^2 + 2x + 3)

3. Find the value of the limit of the function f(x) = \lim_{x \to 0} \frac{\sin(x)}{x}

4. Find the value of the limit of the function f(x) = \lim_{x \to 1} \frac{\cos(x)}{x}

5. Find the value of the limit of the function f(x) = \lim_{x \to \infty} \frac{6x^2 - 5x + 1}{2x^2 + 4}

6. Find the value of the limit of the function f(x) = \lim_{x \to 2} \frac{x^2-16}{x-4}