7. Computing Limits
a.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 \(\lim\limits_{x\to\infty}\dfrac {4+\dfrac{5}{x}-\dfrac{2}{x^2}}{3+\dfrac{4}{x}}\).
We use two column format to give the reasons:
| \(\lim\limits_{x\to\infty}\dfrac{4+\dfrac{5}{x} -\dfrac{2}{x^2}}{3+\dfrac{4}{x}} =\dfrac {\lim\limits_{x\to\infty} \left(4+\dfrac{5}{x}-\dfrac{2}{x^2}\right)} {\lim\limits_{x\to\infty} \left(3+\dfrac{4}{x}\right)}\) |
The limit of a quotient is the quotient of the limits
provided the limit of the denominator is non-zero. |
| \(=\dfrac {\lim\limits_{x\to\infty}4 +\lim\limits_{x\to\infty}\dfrac{5}{x} -\lim\limits_{x\to\infty}\dfrac{2}{x^2}} {\lim\limits_{x\to\infty}3 +\lim\limits_{x\to\infty}\dfrac{4}{x}}\) |
The limit of a sum is the sum of the limits.
The limit of a difference is the difference of the limits. |
| \(=\dfrac {4+5\lim\limits_{x\to\infty}\dfrac{1}{x} -2\lim\limits_{x\to\infty}\dfrac{1}{x^2}} {3+4\lim\limits_{x\to\infty}\dfrac{1}{x}}\) |
The limit of a constant function is the constant.
The limit of a constant times a function is the constant times the limit of the function. |
| \(=\dfrac{4+5\cdot0-2\cdot0}{3+4\cdot0}\) | Special Limit: \(\lim\limits_{x\to\infty}\dfrac{1}{x^p}=0\,\) for \(p=1,2\) |
| \(=\dfrac{4}{3}\) | Simplify |
Notice that the limit of the denominator was \(3\), thereby satisfying the requirement of the quotient rule.
Compute \(\lim\limits_{x\to\infty}\cos\left(\dfrac{\sqrt{2+x^{-3}}}{x}\right)\).
We use two column format to give the reasons:
| \(\lim\limits_{x\to\infty}\cos\left(\dfrac{\sqrt{2+x^{-3}}}{x}\right) =\cos\left(\lim\limits_{x\to\infty}\dfrac{\sqrt{2+x^{-3}}}{x}\right)\) |
The limit of a continuous function of a function is
the continuous function of the limit of the function. |
| \(=\cos\left(\lim\limits_{x\to\infty}\dfrac{1}{x} \lim\limits_{x\to\infty}\sqrt{2+x^{-3}}\right)\) | The limit of a product is the product of the limits. |
| \(=\cos\left(0 \sqrt{\lim\limits_{x\to\infty} (2+x^{-3})}\right)\) |
Special Limit:
\(\lim\limits_{x\to\infty}\dfrac{1}{x^p}=0\) for \(p=1\).
The limit of a power is the power of the limit. |
| Note: We cannot yet conclude the limit is \(\cos(0)\) because we have not yet shown that the square root factor is not \(\infty\) which would give an indeterminate form \(0\cdot\infty\) which requires more work. | |
| \(=\cos\left(0 \sqrt{\lim\limits_{x\to\infty}2 +\lim\limits_{x\to\infty}x^{-3}}\right)\) | The limit of a sum is the sum of the limits. |
| \(=\cos(0 \sqrt{2+0})\) |
The limit of a constant function is the constant.
Special Limit: \(\lim\limits_{x\to\infty}\dfrac{1}{x^p}=0\) for \(p=3\). |
| \(=\cos (0)=1\) | Simplify |