Math 22 Limits

The Limit of a Function

 Definition of the Limit of a Function
 If $f(x)$ becomes arbitrarily close to a single number $L$ as $x$ approaches $c$ from either side, then
 $\lim _{x\to c}f(x)=L$
 which is read as "the limit of $f(x)$ as $x$ approaches $c$ is $L$

Note: Many times the limit of $f(x)$ as $x$ approaches $c$ is simply $f(c)$, so limit can be evaluate by direct substitution as $\lim _{x\to c}f(x)=f(c)$

Properties of Limits

Let $b$ and $c$ be real numbers, let $n$ be a positive integer, and let $f$ and $g$ be functions with the following limits $\lim _{x\to c}f(x)=L$ and $\lim _{x\to c}g(x)=K$. Then

1. Scalar multiple: $\lim _{x\to c}[bf(x)]=bL$

2. Sum or difference: $\lim _{x\to c}[f(x)\pm g(x)]=L\pm K$

3. Product: $\lim _{x\to c}[f(x)\cdot g(x)]=L\cdot K$

4. Quotient: $\lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L}{K}}$

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.

In some case, 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)} can be increase/decrease without bound 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} . We can write 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\infty}

Now, 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}} . By direct substitution, it is of the form 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 \frac {\text{constant}}{0}} , so the answer will be either 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 \infty} or 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 -\infty} . In order to find the limit, we must consider the limit from both side (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^-}} 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 1^+}} ).

When 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\to 1^-} , so 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<1} , hence 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-1<0} . Therefore, 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}=\frac {\text{negative}}{\text{negative}}=\infty}

When 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\to 1^+} , so 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>1} , hence 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-1>0} . Therefore, 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}=\frac {\text{negative}}{\text{positive}}=-\infty}

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