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Calculus Limits

The document explains how to evaluate limits using direct substitution, both for numerical values and infinity. It also discusses methods like factoring, reducing, and multiplying by the conjugate when direct substitution leads to undefined solutions. Additionally, it covers piecewise functions and the concept of right and left-sided limits, noting that differing results indicate that the limit does not exist.

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Philomena Lanuza
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47 views3 pages

Calculus Limits

The document explains how to evaluate limits using direct substitution, both for numerical values and infinity. It also discusses methods like factoring, reducing, and multiplying by the conjugate when direct substitution leads to undefined solutions. Additionally, it covers piecewise functions and the concept of right and left-sided limits, noting that differing results indicate that the limit does not exist.

Uploaded by

Philomena Lanuza
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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• Evaluate by Direct Substitution: Take the value of the limit and evaluate the

function at this value. There are two situations where direct substitution will be
used, direct substitution with a numerical value and direct substitution with
infinity.

Example 1: Limits at a Numerical Value.

2(2)3 + 9 + 3(2) − 1
=
4(2) + 1

√25 + 5 10
=
9 9

Example 2: Limits at Infinity. Limits at infinity are solved by plugging in ∞ or


−∞ into the function for the given variable. When determining limits at infinity,
think more about the trends of the function at infinity rather than the math.

Some Common Trends at Infinity

n a(∞)𝑛 𝑎
=
b(∞) 𝑛
𝑏
𝑛 Where 𝑏 ≠ 0
𝑊her𝑒 𝑛 ≠ 0

(−∞)𝑛 = −∞ (±∞)𝑛 = ∞ (±∞)−𝑛 = 0


𝑊ℎ𝑒𝑛 𝑛 𝑖s 𝑜dd 𝑎nd 𝑛 > 0 When n is even and n > Where n > 0
0
 Factor and Reduce or Multiply by the Conjugate: Use when the first method
yields an undefined solution.
Example 1: Factoring and reducing.

𝑥−4
If 𝑥 were simply evaluated at 4 as shown in the first method, it would yield a zero
in the denominator; therefore, the slope is undefined. One way to avoid this is to
factor the numerator and denominator if applicable.

Reduce by .

Now, if 𝑥 is evaluated at 4, the equation will not yield an undefined slope.

𝑥−4
=4

Example 2: Multiplying by the Conjugate.

Reduce by (𝑥 − 4)

𝑥𝑥 − 4
=4

Example 1: Piecewise functions.

To find the limit as 𝑥 approaches 1 from the left side, the first equation must be
e

used because it defines the function at values less than and equal to one. Since
the equation will not yield an undefined result, direct substitution can be used.

To find the limit as 𝑥 approaches 1 from the right side, the second equation must
be used because it defines the function at values greater than one. Since the
equation will not yield an undefined result, direct substitution can be used.
right sided limits as 𝑥 approaches 1 yields two different answers.
The limit of this function does not exist (DNE) because the values for the left and

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