Limits are a fundamental topic in calculus and often serve as the initial focus in the course. While they may be intimidating to many, their notation and solutions differ significantly from those encountered in previous courses. Fortunately, once we become familiar with their peculiar notation, limits simply represent the idea of describing a function’s behavior as it approaches a certain value. Let’s delve into this intriguing topic!
Limits are typically expressed in the following format: “The limit as x approaches 2 of f(x) is equal to 4.”
$$ \lim_{{x \to 2}} f(x) = 4 $$
Whenever we compute a limit, we are observing the behavior “of” something, often a function of a variable. The variable then “approaches” a specified number along that function. Visually, this implies that, as we traverse a function (denoted as ( f(x) )) towards a destination (here, ( x = 2 )), the function will get arbitrarily “closer” or “approach” a particular output (in this case, a value of 4). The accompanying illustration vividly demonstrates how, as we move along the line left or right towards 2, we approach a height of 4.
Note that we deliberately refrain from asserting whether the function equals 4 when ( x ) is equal to 2. Limits focus on what we expect, not necessarily what we obtain. Even if a function has a gap at a single point, we can still establish a valid limit. In the example below, the limit appears as:
$$\lim_{{x \to 1}} f(x) = 0.5 $$
Despite the absence of an actual value for ( f(1.5) ) due to the gap in the graph, we confidently assert that the limit exists and equals 0.5. Limits represent what we expect, not necessarily what we find. As we approach that gap, it seems like the graph will smoothly continue over it, reaching the value of 0.5. Thus, 0.5 becomes our limit.
In the preceding section, we mentioned “approaching” a value, assuming the approach could be from either direction. However, as we will soon discover, approaching from one direction may yield different results than approaching from another. The notation for a limit approaching “from the right” is denoted by the “+” superscript:
$$\lim_{{x \to a^+}} f(x) $$Invalid response
Similarly, a limit approaching “from the left” is denoted with a “-” superscript:
$$ \lim_{{x \to a^-}} f(x) $$
The following example illustrates these differences.
In this case, the x value of 2 is where these two distinct graphs overlap, making it a perfect scenario to employ left and right limits.
To find the left limit, we trace the curve starting from the left of our x value. As we follow along towards ( x = 2 ) from the left, the curve approaches a y value of 3. Thus,
$$\lim_{{x \to 2^-}} g(x) = 3 $$
Similarly, finding the right limit involves tracing the curve from the right of ( x = 2 ). As we approach ( x = 2 ) from the right, the curve instead approaches a y value of 1. Thus,
$$\lim_{{x \to 2^+}} g(x) = 1 $$
Notice that, in this case, there is no value for our function at ( x = 2 ), since both endpoints are represented by empty circles. However, this does not affect our limits because we are concerned only with what we are approaching, not necessarily what will be there when we arrive. The journey takes precedence over the destination.
While limits from both directions exist and represent valid numbers, the discrepancy between them raises concerns. In cases where the limits from the left and right of a function at a point do not match, we declare that the limit does not exist, indicated as follows:
$$\lim_{{x \to 2^-}} g(x) \quad DNE $$
A limit approaching a value at a vertical asymptote will indefinitely trend upwards or downwards towards infinity. Analyzing the graph of ( \frac{1}{x^2} ) at ( x = 0 ) confirms this.
Thus, we express it as:
$$\lim_{{x \to 0}} f(x) = \infty $$
Similarly, examining the following graph:
reveals that the limit as ( x ) approaches ( a ) is negative infinity:
$$\lim_{{x \to a}} f(x) = -\infty $$
With this, you’ve gained a foundational understanding of handling limits in relation to the graphs of functions. If these concepts still pose challenges or you seek personalized support, the team at Oahu Prep is delighted to offer professional tutoring services. Take care!