Difference between revisions of "Math 22 Limits"
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'''1. Direct Substitution''': Direct Substitution can be used to find the limit of a Polynomial Function. | '''1. Direct Substitution''': Direct Substitution can be used to find the limit of a Polynomial Function. | ||
| − | Example: Evaluate <math>lim_{x\to 3} x^2+2x-1=(3)^2+2(3)-1=14</math> | + | Example: Evaluate <math>\lim_{x\to 3} x^2+2x-1=(3)^2+2(3)-1=14</math> |
'''2. Dividing Out Technique''': When direct substitution fails and numerator or/and denominator can be factored. | '''2. Dividing Out Technique''': When direct substitution fails and numerator or/and denominator can be factored. | ||
| − | Example: Evaluate <math>lim_{x\to 2} \frac {x^2-4}{x^2-x-2}= lim_{x\to 2} \frac {(x-2)(x+2)}{(x-2)(x+1)}=lim_{x\to 2} \frac {x+2}{x+1}</math>. Now we can use direct substitution to get the answer. | + | Example: Evaluate <math>\lim_{x\to 2} \frac {x^2-4}{x^2-x-2}= \lim_{x\to 2} \frac {(x-2)(x+2)}{(x-2)(x+1)}=\lim_{x\to 2} \frac {x+2}{x+1}</math>. Now we can use direct substitution to get the answer. |
'''3. Rationalizing (Using Conjugate)''': When direct substitution fails and either numerator or denominator has a square root. In this case, we can try to multiply both numerator and denominator by the conjugate. | '''3. Rationalizing (Using Conjugate)''': When direct substitution fails and either numerator or denominator has a square root. In this case, we can try to multiply both numerator and denominator by the conjugate. | ||
| − | Example: Evaluate <math>lim_{x\to 0} \frac {\sqrt{x+4}-2}{x}= lim_{x\to 0} \frac {\sqrt{x+4}-2} {x} \cdot \frac {\sqrt{x+4}+2}{\sqrt{x+4}+2}= lim_{x\to 0} \frac {(x+4)-4}{x(\sqrt{x+4}+2)}=lim_{x\to 0} \frac {x}{x(\sqrt{x+4}+2)}=lim_{x\to 0} \frac {1}{\sqrt{x+4}+2}</math>. Now we can use direct substitution to get the answer | + | Example: Evaluate <math>\lim_{x\to 0} \frac {\sqrt{x+4}-2}{x}= \lim_{x\to 0} \frac {\sqrt{x+4}-2} {x} \cdot \frac {\sqrt{x+4}+2}{\sqrt{x+4}+2}= \lim_{x\to 0} \frac {(x+4)-4}{x(\sqrt{x+4}+2)}=\lim_{x\to 0} \frac {x}{x(\sqrt{x+4}+2)}=\lim_{x\to 0} \frac {1}{\sqrt{x+4}+2}</math>. Now we can use direct substitution to get the answer |
==One-Sided Limits and Unbounded Function== | ==One-Sided Limits and Unbounded Function== | ||
Revision as of 06:40, 14 July 2020
The Limit of a Function
Definition of the Limit of a Function If becomes arbitrarily close to a single number as approaches from either side, then which is read as "the limit of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L}
Note: Many times the limit of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is simply Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(c)} , so limit can be evaluate by direct substitution as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c} f(x)=f(c)}
Properties of Limits
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} be real numbers, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} be a positive integer, and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} be functions with the following limits Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c} f(x)=L} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c} g(x)=K} . Then
1. Scalar multiple: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c} [bf(x)]=bL}
2. Sum or difference: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c} [f(x)\pm g(x)]=L\pm K}
3. Product: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c} [f(x)\cdot g(x)]=L\cdot K}
4. Quotient: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c} \frac {f(x)}{g(x)}=\frac {L}{K}}
5. Power: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c} [f(x)]^n=L^n}
6. Radical: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c} \sqrt[n]{f(x)}=\sqrt[n]{L}}
Techniques for Evaluating Limits
1. Direct Substitution: Direct Substitution can be used to find the limit of a Polynomial Function.
Example: Evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to 3} x^2+2x-1=(3)^2+2(3)-1=14}
2. Dividing Out Technique: When direct substitution fails and numerator or/and denominator can be factored.
Example: Evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to 2} \frac {x^2-4}{x^2-x-2}= \lim_{x\to 2} \frac {(x-2)(x+2)}{(x-2)(x+1)}=\lim_{x\to 2} \frac {x+2}{x+1}} . Now we can use direct substitution to get the answer.
3. Rationalizing (Using Conjugate): When direct substitution fails and either numerator or denominator has a square root. In this case, we can try to multiply both numerator and denominator by the conjugate.
Example: Evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to 0} \frac {\sqrt{x+4}-2}{x}= \lim_{x\to 0} \frac {\sqrt{x+4}-2} {x} \cdot \frac {\sqrt{x+4}+2}{\sqrt{x+4}+2}= \lim_{x\to 0} \frac {(x+4)-4}{x(\sqrt{x+4}+2)}=\lim_{x\to 0} \frac {x}{x(\sqrt{x+4}+2)}=\lim_{x\to 0} \frac {1}{\sqrt{x+4}+2}} . Now we can use direct substitution to get the answer
One-Sided Limits and Unbounded Function
when a function approaches a different value from the left of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}
than it approaches from the right of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}
, the limit does not exists. However, this type of behavior can be described more concisely with
the concept of a one-sided limit. We denote
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c^{-}} f(x)=L}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c^{+}} f(x)=K}
One-sided Limit is related to unbounded function.
Consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to 1} \frac {2}{x-1}}
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