MA441-Lecture 4

MA441- LEC 4 - 1.6 Calculating limits using the limit Laws

Goal

Limit Laws and their restrictions

Limit Laws 1-5

Suppose \(c\) is a constant and the limits \[\lim_{x\to a} f(x)~and~\lim_{x\to a} g(x)\] exist. Then

\(\displaystyle\lim_{x\to a} [f(x)+g(x)]=\lim_{x\to a} f(x)+\lim_{x\to a} g(x)\)
\(\displaystyle\lim_{x\to a} [f(x)-g(x)]=\lim_{x\to a} f(x)-\lim_{x\to a} g(x)\)
\(\displaystyle\lim_{x\to a} [ c\cdot g(x)]=c \cdot \lim_{x\to a} g(x)\)
\(\displaystyle\lim_{x\to a} [f(x) \cdot g(x)]=\lim_{x\to a} f(x) \cdot \lim_{x\to a} g(x)\)
\(\displaystyle\lim_{x\to a} \frac{f(x)}{g(x)}=\frac{\ds \lim_{x\to a} f(x)}{\ds \lim_{x\to a} g(x)}~if~\lim_{x\to a} g(x)\neq0\)

Remark

Do not forget \(\displaystyle\lim_{x\to a} g(x)\neq0\) for law 5.

\[\displaystyle\lim_{x\to a} \frac{f(x)}{g(x)}=\frac{\ds \lim_{x\to a} f(x)}{\ds \lim_{x\to a} g(x)}~if~\lim_{x\to a} g(x)\neq0\]

One-sided limits also have similar rules.

Example 1

Using the Limit Laws and the graphs of \(f\) and \(g\) to evaluate the following limits, if they exists. If it does not exists, check the infinite limits.

\(\displaystyle\lim_{x\to -2} [f(x)+5g(x)]\), \(\displaystyle\lim_{x\to 1} [f(x)g(x)]\)
\(\displaystyle\lim_{x\to 2^-} \frac{f(x)}{g(x)},~\lim_{x\to 2^+} \frac{f(x)}{g(x)}\)

For (c).

Since \(\lim_{x\to 2} f(x)\approx 1.5\neq0\) and \( \lim_{x\to 2} g(x)=0\),

it may give an infinite limit. So we need to check one-sided limits.

If \( x ~< 2 \) and \( x \) is very close to \(2\), \(f(x)~>0,~g(x)<0\).

\[ \lim_{x\to 2^-} \frac{f(x)}{g(x)}=-\infty \]

If \(x>2\) and \(x\) is very close to \( 2\), \(f(x)>0,~g(x)>0\). \[ \lim_{x\to 2^+} \frac{f(x)}{g(x)}=+\infty \]

Example 2.

Let \(\displaystyle\lim_{x\to 1^-} f(x)=2,~\lim_{x\to 1^+} f(x)=-1\) \(,~\lim_{x\to 1^-} g(x)=-3, ~\lim_{x\to 1^+} g(x)=0\).

Find \[\displaystyle\lim_{x\to 1} [f(x)+g(x)]\]

Limit Laws 6-11

6. \(\displaystyle\lim_{x\to a} [f(x)]^n=[\lim_{x\to a} f(x)]^n\) when \(n\) is a positive integer.
7. \(\displaystyle\lim_{x\to a} c=c\)
8. \(\displaystyle\lim_{x\to a} x=a\)
9. \(\displaystyle\lim_{x\to a} x^n =a^n\) when \(n\) is a positive integer.
10. \(\displaystyle\lim_{x\to a} \sqrt[n]{x}=\sqrt[n]{a}\) when \(n\) is a positive integer.

[ If \(n\) is even, we assume \(a>0\)]

11. \(\displaystyle\lim_{x\to a} \sqrt[n]{f(x)}=\sqrt[n]{\lim_{x\to a} f(x)}\) when \(n\) is a positive integer.

[ If \(n\) is even, we assume \(\lim_{x\to a} f(x) ~>0\)]

Direct substitution Property

Let \(f\) is a polynomial or a rational function or radical function.

If a is in the interior of domain of \(f\),

\[ \lim_ {x\to a} f(x)=f(a) \]

If \(a\) is in the boundary of the domain,

\[ \lim_ {x\to a^{\pm}} f(x)=f(a) \] depending on the side of boundary.

(Namely, \(f(x)\) is continuous on the domain.)

Ex. \(f(x)=\sqrt{x}\) and \(Domain(f)=[0,\infty)\)

\[ \lim_ {x\to a}f(x)=f(0)=0,~for~a>0\]

\[ \lim_ {x\to 0^{+}}f(x)=f(0)=0\]

Example 3

\[ \lim_ {x\to 5} (2x^2-3x+4) \]

\[ \lim_ {x\to -2} \frac{x^3+2x^2-1}{5-3x} \]

Limit doesn't depends on finite many values.

If \( f(x)=g(x) \) when \( x\neq a\), then

\[ \lim_ {x \to a} f(x)=\lim_{x\to a} g(x) \]

if the limits exist.

Example 4.

\[ \lim_ {h\to 0} \frac{(3+h)^2-9}{h} \]

Example 5.

\[ \lim_ {t\to 0} \frac{\sqrt{t^2+9}-3}{t^2} \]