Introduction to Limits and Derivatives
In Mathematics, a limit is defined as a value that a function approaches as the input approaches some value. Limits are important in calculus and mathematical analysis and used to define integrals, derivatives, and continuity. Take a look at the basics of limits and derivatives in this article.
To express the limit of a function, we represent it as:
Properties of Limits
Let p and q be two functions and a be a value such that exists
Limits and Derivatives
A derivative refers to the instantaneous rate of change of a quantity with respect to the other. It helps to investigate the moment by moment nature of an amount. Limits and derivatives are tied closely by the definition itself. The Derivative of a function is represented as:
For the function f, its derivative is said to be f'(x) given the equation above exists.
Properties of Derivatives
This varying rate of change is termed as a Derivative.
The derivative is primarily used when there is some varying quantity, and the rate of change is not constant. A Derivative is used to measure the sensitivity of one variable (dependent variable) with respect to another variable (independent variable).
A derivative refers to the instantaneous rate of change of a quantity with respect to the other. It helps to investigate the moment by moment nature of an amount.
Let a car takes ‘t’ seconds to move from a point ‘a’ to ’b’.
But how long will it take to move from point ‘a’ to ‘c’? or How much distance will it cover in ‘t-1’ seconds?
This can be known from the velocity that is as follows:
Velocity (v) = d(x)/d(t)
Where ‘x’ is the distance travelled and ‘t’ is the time taken to cover that distance.
This will give you the distance covered per unit time so that we can analyze any distance covered in any interval of time.
Since the very definition of derivatives involves limits in a rather direct fashion, we expect the rules of derivatives to follow closely that of limits as given below:
Steps to find Derivative:
- Change x by the smallest possible value and let that be (dx) and so the function becomes f(x+dx)
- Get the change in value of function that is : f(x + dx) – f(x)
- The rate of change in function f(x) on changing from ‘x’ to ‘x+dx’ will be f(x + dx) – f(x)d(x)
Now d(x) is ignorable because it is considered to be too small.
Let f(x) be a function where f(x) = x2
The derivative of x2 is 2x means that with every unit change in x, the value of the function becomes twice (2x).
Limits and Derivatives
When dx is made so small that is becoming almost nothing. With Limits, we mean to say that X approaches zero but does not become zero.
Mathematically: means for all real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |f(x) − L| < ε
- To differentiate a power of x that is in the denominator, first express it as a power with a negative exponent. Eg. 1/x2 = x−2
- Derivative rules simplify the process of differentiating polynomial functions.
- To differentiate a radical, first, express it as a power with a rational exponent
Example 1: To Compute limx→−4 (5x2 + 8x – 3)
Solution: First, use property 2 to divide the limit into three separate limits. Then use property 1 to bring the constants out of the first two. This gives,
limx→−4(5x2 + 8x − 3) = limx→−4(5x2) + limx→−4(8x) − limx→−4(3)
= 5(−4)2 + 8(−4) − 3
= 80 – 32 – 3 = 45
Example 2: To Compute limx→6[(x−3)(x−2)x−4]
Example 3: A skydiver jumps out of a plane from a height of 2200 m. The skydiver’s height above the ground, in meters, after t seconds is represented by the function h(t) = 2200 − 4.9t2 (assuming air resistance is not a factor). How fast is the skydiver falling after 4 s?
Solution: The instantaneous rate of change of the height of the skydiver at any point in time is represented by the derivative of the height function.
h(t) = 2200−4.9t2
h′(t) = 0 − 4.9(2t) = −9.8t
Substitute t=4 into the derivative function to find the instantaneous rate of change at 4 s.
Therefore, after 4 s, the skydiver is falling at a rate of 39.2 m/s.