Limits And Derivatives
Introduction to limits
Calculus: Calculus is that branch of mathematics which mainly deals with the study of change in the value of a function as the points in the domain change.
Goals of Calculus
- To find tangent (slope) at a point on curve.
- To find area under the curve.
\(f(x)=x+2\)
\(f(2)=2+2=4\)\(f(x)=\dfrac{x^2-4}{x-2}\)
\(f(2)=\dfrac{2^2-4}{2-2}=\frac{0}{0}=undefined\)
At \(x=2\) value of following functions are
on factorization
\(f(x)=\dfrac{x^2-4}{x-2}=\dfrac{(x+2)(x-2)}{x-2}=x+2\)
On factorization above function follow the case first and its value is 4. But as per case 2 the value of function at 2 is undefined.
We can write the case 2nd in limit form
\(\lim\limits_{x \to 2} \dfrac{x^2-4}{x-2}=4\)
In general as x → a, f (x) → l, then l is called limit of the function f (x) which is symbolically written as \(\lim\limits_{x \to a} f(x)=l\)
Idea of Limit
\(f(x)=\dfrac{x^2-4}{x-2}\)
| \(x\) | 1.5 | 1.9 | 1.99 | 1.999 | 2 | 2.001 | 2.01 | 2.1 | 2.5 |
|---|---|---|---|---|---|---|---|---|---|
| \(f(x)\) | 3.5 | 3.9 | 3.99 | 3.999 | undefined | 4.001 | 4.01 | 4.1 | 4.5 |
Left hand limit \(\lim\limits_{x \to 2^-} f(x)=4\)
Right hand limit \(\lim\limits_{x \to 2^+} f(x)=4\)
\(\lim\limits_{x \to 2^-} f(x) =\lim\limits_{x \to 2^+} f(x)\)
When left hand limit and right hand limits are equal then it can be written as \(\lim\limits_{x \to 2} f(x)=4\)
we can say that x is approaches to or tends to 2, but not equal to 2.
\(f(x)=\dfrac{|x|}{x}\)
| \(x\) | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|---|
| \(f(x)\) | -2 | -2 | -2 | -2 | undefined | 2 | 2 | 2 | 2 |
Left hand limit \(\lim\limits_{x \to 0^-} f(x)=-2\)
Right hand limit \(\lim\limits_{x \to 0^+} f(x)=2\)
\(\lim\limits_{x \to 0^-} f(x) \neq \lim\limits_{x \to 0^+} f(x)\)
When left hand limit and right hand limits are not equal then \(\lim\limits_{x \to 0} f(x)\) does not exist.
\(\lim\limits_{x \to a^-} f(x)\) is the expected value of f at x = a given the values of f near x to the left of a. This value is called the left hand limit of f at a.
\(\lim\limits_{x \to a^+} f(x)\) is the expected value f at x = a given the values of f near x to the right of a. This value is called the right hand limit of f(x) at a.
\(\lim\limits_{x \to a} f(x)\) the right and left hand limits coincide, we call that common value as the limit of f(x) at x = a.
Algebra of Limits
Let f(x) ang g(x) be two functions such that \(\lim\limits_{x \to a} f(x)\) and \(\lim\limits_{x \to a} g(x)\) exist. Then
- \(\lim\limits_{x \to a} [f(x) + g(x)]=\lim\limits_{x \to a} f(x)+\lim\limits_{x \to a} g(x)\)
- \(\lim\limits_{x \to a} [f(x) - g(x)]=\lim\limits_{x \to a} f(x)-\lim\limits_{x \to a} g(x)\)
- \(\lim\limits_{x \to a} [f(x) . g(x)]=\lim\limits_{x \to a} f(x).\lim\limits_{x \to a} g(x)\)
- \(\lim\limits_{x \to a} \dfrac{f(x)}{g(x)}=\dfrac{\lim\limits_{x \to a} f(x)}{\lim\limits_{x \to a} g(x)}\)
- \(\lim\limits_{x \to a} [k . f(x)]=k.\lim\limits_{x \to a} f(x)\)
where k is a constant.