Solved Questions on Estimating Limits from Graphs

Question 1: Find the 

Answer: 

Graph of x² is an upward parabola which is centered at origin. The figure below shows the graph of the given function. 

Notice that the function starts moving towards taking a value of 1 as one moves towards the value of x = 1. 

Thus, 

Question 2: Find the 

Answer: 

Graph of log(x) is a saturating function. The figure below shows the graph of the given function. 

Notice that the function starts moving towards taking a value of -∞ as one moves towards the value of x = 0. This is an example of unbounded limit mentioned above.  

Thus, 

Question 3: Find the limit of the function at x = 1. 

Solution: 

In the graph, the white dot at x = 1 indicates that the function is not defined at x = 1. That means, there should be no value of the function at x =1. 

Limit allows us to calculate the values the function was approaching had it been defined at x =1. 

So, in this case from graph it can be seen that the function is approaching value of 5. 

Question 4: Find the value of the limit of the function f(x) = at x = 0. 

Solution:

The figure shows the graph for the given function

The figure makes it clear that there are two limits depending on from which side we are approaching the function. 

Left Hand Side Limit: 

While approaching zero from the negative side of the origin takes the function to negative infinity. 

Right Hand Side Limit: 

While approaching zero from the positive side of the origin takes the function to positive infinity. 

Question 5: Find the 

Answer: 

Graph of x² is an upward parabola which is centered at origin. The figure below shows the graph of the given function. 

Notice that the function starts moving towards taking a value of 1 as one moves towards the value of x = 1. 

Thus, 

Question 5: Find out the limit at x = 0 for the given function, 

Answer: 

In the figure, the given function is not continuous at origin. That means there will be two different values of the limit – one from left-hand side and another from right-hand side. 

Left- Hand Limit for the function 

Right – Hand Limit for the function, 

The concept of limits has been around for thousands of years. Earlier mathematicians in ancient civilizations used limits to approximate the area of a circle. But the formal concept was not around till the 19th century. This concept is essential to calculus and serves as a building block for analyzing derivatives, continuity, and differentiability. Intuitively, limits give us an idea about the values function approaches at a particular value of x. Using this idea, limits can also be estimated to a certain extent just by looking at the graph. Let’s look at these ideas in detail.