Limit laws

Limit laws

Basic concepts: Function notation,

Lessons

Here are some properties of limits:

1) limxax=a\lim_{x \to a} x = a
2) limxac=c\lim_{x \to a} c = c
3) limxa[cf(x)]=climxaf(x)\lim_{x \to a} [cf(x)] = c\lim_{x \to a}f(x)
4) limxa[f(x)±g(x)]=limxaf(x)±limxag(x)\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a}f(x) \pm \lim_{x \to a}g(x)
5) limxa[f(x)g(x)]=limxaf(x)limxag(x)\lim_{x \to a} [f(x) g(x)] = \lim_{x \to a}f(x) \lim_{x \to a}g(x)
6) limxaf(x)g(x)=limxaf(x)limxag(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 limxag(x)0\lim_{x \to a}g(x) \neq0
7) limxa[f(x)]n=[limxaf(x)]n\lim_{x \to a} [f(x)]^n=[\lim_{x \to a}f(x)]^n

Where c is a constant, limxaf(x)\lim_{x \to a} f(x) and limxag(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
limxaP(x)=P(a)\lim_{x \to a} P(x)=P(a)
  • 1.
    Evaluating Limits of Functions
    Evaluate the following limits using the property of limits:
    a)
    limx2x2+4x+3\lim_{x \to 2} x^2+4x+3

    b)
    limx23(x2+4x+3)2\lim_{x \to 2} 3(x^2+4x+3)^2

    c)
    limx123x+4x22+x4\lim_{x \to 1} \frac{2-3x+4x^2}{2+x^4}

    d)
    limx04(3)x\lim_{x \to 0} 4(3)^x

    e)
    limxπ23(sinx)4\lim_{x \to \frac{\pi}{2}} 3(\sin x)^4


  • 2.
    Evaluating Limits with specific limits given
    Given that limx5f(x)=3\lim_{x \to 5} f(x)=-3, limx5g(x)=5\lim_{x \to 5} g(x)=5, limx5h(x)=2\lim_{x \to 5} h(x)=2, use the limit properties to compute the following limits:
    a)
    limx5[5f(x)2g(x)]\lim_{x \to 5} [5f(x)-2g(x)]

    b)
    limx5[g(x)f(x)+3h(x)]\lim_{x \to 5} [g(x)f(x)+3h(x)]

    c)
    limx52g(x)h(x)\lim_{x \to 5} \frac{2g(x)}{h(x)}

    d)
    limx55[f(x)]3g(x)\lim_{x \to 5} \frac{5[f(x)]^3}{g(x)}