7. Computing Limits
a. Limits at Finite Numbers
2. Limit Laws
a. Examples
In general, the process of finding the limit of a function is to repeatedly apply the Limit Laws until we get down to the Special Limits. If there is a condition to check, assume it holds and check it at the end.
Compute \(\lim\limits_{x\to2}\dfrac{(3x^2+2)^2(2x-1)}{x+4}\).
We use two column format to give the reasons:
| \(\lim\limits_{x\to2}\dfrac{(3x^2+2)^2(2x-1)}{x+4}\) | |
| \(=\dfrac {\lim\limits_{x\to2}\left[(3x^2+2)^2(2x-1)\right]} {\lim\limits_{x\to2}(x+4)}\) | The limit of a quotient is the quotient of the limits provided the limit of the denominator is non-zero. |
| \(=\dfrac {\left[\lim\limits_{x\to2}(3x^2+2)\right]^2 \lim\limits_{x\to2}(2x-1)} {\lim\limits_{x\to2}(x+4)}\) |
The limit of a product is the product of the limits.
The limit of a power is the power of the limit. |
| \(=\dfrac {\left(\lim\limits_{x\to2}3x^2+\lim\limits_{x\to2}2\right)^2 (\lim\limits_{x\to2}2x-\lim\limits_{x\to2}1)} {\lim\limits_{x\to2}x+\lim\limits_{x\to2}4}\) |
The limit of a sum is the sum of the limits.
The limit of a difference is the difference of the limits. |
| \(=\dfrac {\left(3\lim\limits_{x\to2}x^2+2\right)^2 (2\lim\limits_{x\to2}x-1)} {\lim\limits_{x\to2}x+4}\) |
The limit of a constant times a function is the
constant times the limit of the function.
The limit of a constant function is the constant. |
| \(=\dfrac {(3\cdot2^2+2)^2(2\cdot2-1)} {2+4}\) | Special Limit: \(\lim\limits_{x\to2}x^p=2^p\,\) for \(p=1,2\) |
| \(=\dfrac{14^2\cdot3}{6}=98\) | Simplify |
Notice that the limit of the denominator was \(6\), thereby satisfying the requirement of the quotient rule.
Compute \(\lim\limits_{x\to3}\arcsin\left(\dfrac{\sqrt{(x+1)^2+x^2}}{x^2+1}\right)\).
\(\lim\limits_{x\to3}\arcsin\left(\dfrac{\sqrt{(x+1)^2+x^2}}{x^2+1}\right) =\dfrac{\pi}{6}\)
We use two column format to give the reasons:
| \(\lim\limits_{x\to3}\arcsin\left(\dfrac{\sqrt{(x+1)^2+x^2}}{x^2+1}\right)\) | |
| \(=\arcsin\lim\limits_{x\to3}\left(\dfrac{\sqrt{(x+1)^2+x^2}}{x^2+1}\right)\) | Continuous functions pull out of limits. |
| \(=\arcsin\dfrac{\lim\limits_{x\to3}\sqrt{(x+1)^2+x^2}} {\lim\limits_{x\to3}(x^2+1)}\) | The limit of a quotient is the quotient of the limits provided the limit of the denominator is non-zero. |
| \(=\arcsin\dfrac{\sqrt{\lim\limits_{x\to3}\left[(x+1)^2+x^2\right]}} {\lim\limits_{x\to3}(x^2+1)}\) | The limit of a power is the power of the limit. |
| \(=\arcsin\dfrac{\sqrt{\left(\lim\limits_{x\to3}x+\lim\limits_{x\to3}1\right)^2+\lim\limits_{x\to3}x^2}} {\lim\limits_{x\to3}x^2+\lim\limits_{x\to3}1}\) |
The limit of a sum is the sum of the limits.
The limit of a power is the power of the limit. |
| \(=\arcsin\dfrac{\sqrt{\left(3+1\right)^2+3^2}}{3^2+1}\) |
The limit of a constant is the constant.
Special Limit: \(\lim\limits_{x\to3}x^p=3^p\) for \(p=1,2\). |
| \(=\arcsin\dfrac{5}{10}=\arcsin\dfrac{1}{2}=\dfrac{\pi}{6}\) | Simplify |
Notice that the limit of the denominator was \(10\), thereby satisfying the requirement of the quotient rule.