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In this blog post, we will discuss limits, continuity, and differentiability. Limits are used to determine the behavior of a function near a certain point. continuity is used to determine whether or not a function is smooth near a certain point. differentiability is used to determine how much a function changes at a certain point. We will also look at some examples to help illustrate these concepts.
What Are Limits, Continuity, and differentiability?
Limits, continuity, and differentiability are important concepts in mathematics. Limits are used to determine the behavior of a function near a certain point. Continuity is used to determine whether or not a function is smooth near a certain point. Differentiability is used to determine how much a function changes at a certain point. In this blog post, we will look at some examples that illustrate these ideas and discuss how they are used in mathematics. JEE Main Mathematics Limit continuity and differentiability are important concepts in calculus.
What is a Limit?
A limit is the value that a function approaches as the input gets closer and closer to a certain number. For instance, if you graphed the function y=x² on a coordinate plane, the limit of y as x approaches 0 would be equal to the square root of 0, or simply 0. This means that no matter how close you get to x=0 on your graph, the value of y will always be just shy of 0.
There are also limits at infinity- these represent values that a function either takes on or approaches infinitely as it gets larger and larger. A limit is used to determine the behavior of a function near a certain point.
The limit of a function, f(x), as x approaches c is denoted by lim f(x) = L. This means that the limit of the function as x approaches c is equal to L. We can also write this as:
f(x)→L
or
lim f(x) = L if and only if for every ε > 0 there exists a corresponding N such that for all x with |x-c|≤N , we have |f(x)-L|≤ε .
What is Continuity?
A function f(x) is continuous if the following three statements are true:
The function exists at c, i.e. lim f(x) = L (this must be true for every point that we want to show that it is continuous). The function has a finite value at c, i.e. L ≠ ∞ . For any ε > 0 there exists an N such that |f(x)-L|≤ε whenever |x-c| ≤ N . Or equivalently: lim f(x)=f(c). This means that the left side of this limit evaluated at c is equal to the right side of this limit evaluated at c.
f(x)→L
or
lim f(x)=L, when x →c if and only if: lim f(x)=f(c). This means that the left side of this limit evaluated at c is equal to the right side of this limit evaluated at c.
What is Differentiability?
A function f (x) is differentiable in a point p as long as its derivative exists there, i.e.: lim [f (p+h)-f (p)]/h = l . If a function has a derivative in that point then we say it’s “differentiable in p”, if not we say it’s “not differentiable in p”.
It is important to note that the derivative of a function doesn’t have to be defined everywhere. The functions below are examples of such cases:
f (x) = |x|, f (0)=cos(π/12) and f (0)=sin(π/12) both do not exist yet their derivatives at 0 still exist because they follow from the definition: I’m [f (p+h)-f (p)]/h . If you try to calculate this limit you will get an indeterminate form: 0∞ which means that we can use L’Hospital’s rule to calculate it.
f (x) = x^n for any n≠0 is not differentiable at 0 because the derivative doesn’t exist when x=0. However, if we take the derivative of both sides with respect to h, we get: I’m [h(x+Δh)-h(x)]/Δh which does exist when h≠0. So, even though the function itself isn’t differentiable at a certain point, its derivative is.
Many functions are continuous but not differentiable and vice versa. A good example of a continuous but not differentiable function is the absolute value function: |x|. It’s easy to see that this function is continuous at all real numbers because the left and right limits of it coincide at every point.
Limits continuity and differentiability problems
To see whether or not a function is continuous, it’s important to know these four rules:
- The sum rule states that if f and g are both continuous functions then so is their sum h(x)=f(x)+g(x). The product rule states that if f and g are both continuous functions then so is their product h(x)=f(g).
- The quotient rule states that if f and g are functions such that g is continuous at x=a and not equal to zero, then h(x)=f/g is continuous at x=a provided f(a)/g(a)0.
- The constant multiple rules say that if f is a continuous function, then so is cf for any real number c.
- The composition rule states that if f and g are functions such that g is continuous at x=a and f is continuous at g(x)=a, then the function h(x)=f[g(x)]is also continuous.
Conclusion
Limits, continuity, and differentiability are used in physics, chemistry, and biology to solve problems involving forces acting on objects. It can be applied to chemical reactions by studying how reactants change over time or how products form from their constituents at different concentrations.
