Limits and Continuity – Properties of Limits

The general properties of the limits are given below:

Let’s do some examples to demonstrate the limit properties.

Example 1: If f(x) = x2+5x+6, g(x) = x2-3x+4, and h(x) = x2-2x-4 find the following:

Solution:

In this case, the result inside the square root was positive. Therefore, we could take the square root. We cannot take the square root of a negative number (technically you could, but you get an imaginary solution). However, for this section, we will not consider imaginary limits.

Example 2: Evaluate the following limits using the graph of f(x) below.

Finding Limits by Direct Substitution

In some cases, you can find the limits of functions simply by substituting x = c inside the function and evaluating f(c). In other cases, direct substitution will not work. If direct substitution does not work, then you need to utilize other methods (such as factoring, multiplying by conjugates, and the squeeze theorem) to find the limit at x = c. Those methods will be discussed in later sections.

Example 3: Evaluate the following limits:

Solution:

Since this is a polynomial function, we can evaluate the limit of any x value by direct substitution.

Since this is a polynomial function, we can evaluate the limit of any x value by direct substitution.

Since this is a rational function, when doing direct substitution, we need to make sure that the denominator does not equal zero. If it does, then the limit is undefined and we must use another method to find the limit.

Since the function is a constant, the value of the limit at any x value will be 3.

References:
https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-2/e/limits-intro
http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx
https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/SandS/lHopital/define_limit.html
https://www.toppr.com/guides/maths/limits-and-derivatives/limits/
https://www.chegg.com/homework-help/questions-and-answers/based-graph-f-x-figure-determine-f-x-q30350642
http://scidiv.bellevuecollege.edu/dh/Calculus_all/CC_1_2_LimitProps

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