Understanding the Limit of a Function

The limit of a function refers to the value that the function approaches as x approaches a given value.

For instance, let’s consider the function f(x)=x+2.

To find the limit of f(x) as x approaches 5, we can manually calculate the value of f(x) as x gets closer and closer to 5.

There are two ways for x to approach 5, from a number smaller than 5 (we say that x approaches 5 from the left) or from a number greater than 5 (we say that x approaches 5 from the right).

When x approaches from the left, it gets bigger and bigger but does not exceed 5. In contrast, when it approaches from the right, it gets smaller and smaller but does not go below 5.

The table below shows that values of f(x) as x approaches 5 from the left:

We see that as x approaches 5 from the left, f(x) approaches 7. Mathematically, we denote this as

\lim_{x\to5^-}(x+2)=7

5^- indicates that x approaches 5 from the left. Hence, this limit is known as the left-hand limit.

Next, let’s see how the value of f(x) changes as x approaches 5 from the right:

Similarly, as x approaches 5 from the right, f(x) approaches 7. Mathematically, we denote this as

\lim_{x\to5^+}(x+2)=7

5^+ indicates that x approaches 5 from the right. Hence, this limit is known as the right-hand limit.

As both the right-hand and left-hand limits exist and are equal to each other, we say that the general limit exists. Mathematically, we denote this as

\lim_{x\to5}(x+2)=7

In the preceding example, the general limit of x+2 as x approaches 5 (mathematically, we say that as x tends to 5) exists, and can be found by simply substituting x=5 into x+2.

In other words,

\lim_{x\to5}(x+2)=5+2=7

However, this is not always the case. Sometimes, alternative techniques like factoring, rationalization, utilizing limit properties, and employing L’Hopital’s rule might be necessary to determine the limit of a function.

The comprehensive process of finding limits involves various strategies that extend beyond the coverage of this post.

For instance, let’s consider the function f(x)=\frac{1}{x-2}. The table below shows how f(x) behaves as x approaches 2 from the left and the right.

Notice that as x approaches 2 from the left, f(x) gets more and more negative (i.e., it is a negative number that gets smaller and smaller).

In fact, if you use a number for x that is extremely close to, but smaller than 2 (say 1.99999999999), you’ll get a negative number for f(x) that is so small that you either get a “divide by zero” error, or negative infinity as the answer.

Therefore, we say that as x approaches 2 from the left, f(x) approaches negative infinity:

\lim_{x\to2^-}\frac{1}{x-2}=-\infin

Next, observe that as x approaches 2 from the right, f(x) gets more and more positive (i.e., it is a positive number that gets bigger and bigger).

If you use a number for x that is extremely close to, but greater than 2 (say 2.0000000000001), you’ll get a positive number for f(x) that is so huge that the calculator either gives you a “divide by zero” error, or tells you that the answer is infinity.

Therefore, we say that as x approaches 2 from the right, f(x) approaches positive infinity:

\lim_{x\to2^+}\frac{1}{x-2}=\infin

As the left-hand and right-hand limits are not equal to each other in this example, the general limit for \frac{1}{x-2} as x approaches 2 does not exist.

Limits and Calculus

Limits are a fundamental concept in calculus. They enable us to calculate the instantaneous rate of change of a function and provide the foundation for deriving the rules used to find derivatives and integrals.

The video below provides an excellent explanation of how we can use limits to determine the gradient of a curve and derive the power rule for differentiation.

Do check it out: