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  1. Limit Laws Overview:
    7 Properties of Limit Laws
  1. Evaluating Limits of Functions
    Evaluate the following limits using the property of limits:
    1. limโกxโ†’2x2+4x+3\lim_{x \to 2} x^2+4x+3
    2. limโกxโ†’23(x2+4x+3)2\lim_{x \to 2} 3(x^2+4x+3)^2
    3. limโกxโ†’12โˆ’3x+4x22+x4\lim_{x \to 1} \frac{2-3x+4x^2}{2+x^4}
    4. limโกxโ†’04(3)x\lim_{x \to 0} 4(3)^x
    5. limโกxโ†’ฯ€23(sinโกx)4\lim_{x \to \frac{\pi}{2}} 3(\sin x)^4
  2. Evaluating Limits with specific limits given
    Given that limโกxโ†’5f(x)=โˆ’3\lim_{x \to 5} f(x)=-3, limโกxโ†’5g(x)=5\lim_{x \to 5} g(x)=5, limโกxโ†’5h(x)=2\lim_{x \to 5} h(x)=2, use the limit properties to compute the following limits:
    1. limโกxโ†’5[5f(x)โˆ’2g(x)]\lim_{x \to 5} [5f(x)-2g(x)]
    2. limโกxโ†’5[g(x)f(x)+3h(x)]\lim_{x \to 5} [g(x)f(x)+3h(x)]
    3. limโกxโ†’52g(x)h(x)\lim_{x \to 5} \frac{2g(x)}{h(x)}
    4. limโกxโ†’55[f(x)]3g(x)\lim_{x \to 5} \frac{5[f(x)]^3}{g(x)}
Topic Notes
Here are some properties of limits:

1) limโกxโ†’ax=a\lim_{x \to a} x = a
2) limโกxโ†’ac=c\lim_{x \to a} c = c
3) limโกxโ†’a[cf(x)]=climโกxโ†’af(x)\lim_{x \to a} [cf(x)] = c\lim_{x \to a}f(x)
4) limโกxโ†’a[f(x)ยฑg(x)]=limโกxโ†’af(x)ยฑlimโกxโ†’ag(x)\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a}f(x) \pm \lim_{x \to a}g(x)
5) limโกxโ†’a[f(x)g(x)]=limโกxโ†’af(x)limโกxโ†’ag(x)\lim_{x \to a} [f(x) g(x)] = \lim_{x \to a}f(x) \lim_{x \to a}g(x)
6) limโกxโ†’af(x)g(x)=limโกxโ†’af(x)limโกxโ†’ag(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)}, only if limโกxโ†’ag(x)โ‰ 0\lim_{x \to a}g(x) \neq0
7) limโกxโ†’a[f(x)]n=[limโกxโ†’af(x)]n\lim_{x \to a} [f(x)]^n=[\lim_{x \to a}f(x)]^n

Where c is a constant, limโกxโ†’af(x)\lim_{x \to a} f(x) and limโกxโ†’ag(x)\lim_{x \to a} g(x) exist.

Here is a fact that may be useful to you.
If P(x)P(x) is a polynomial, then
limโกxโ†’aP(x)=P(a)\lim_{x \to a} P(x)=P(a)
Basic Concepts