Master the concept of Limits in CBSE Class 11 Applied Mathematics. Learn to evaluate limits of algebraic functions with step-by-step NCERT-style solutions.
A limit describes the value that a function approaches as the input approaches a specific point. We write this as lim x→a f(x) = L, meaning that as x gets closer to a, f(x) gets arbitrarily close to L. It is important to note that the function does not need to be defined at x = a for the limit to exist.
Step 1: Identify the function f(x) = x² + 3 and the point a = 2.
Step 2: Substitute x = 2 directly into the expression.
Step 3: Calculate (2)² + 3 = 4 + 3 = 7.
Answer: 7
Limits follow specific algebraic rules that allow us to simplify complex expressions. If lim x→a f(x) = L and lim x→a g(x) = M, then the limit of the sum, difference, product, and quotient can be calculated individually. These properties are essential for solving polynomial and rational functions.
Step 1: Apply the sum property: lim x→3 2x + lim x→3 x².
Step 2: Calculate individual limits: 2(3) = 6 and (3)² = 9.
Step 3: Add the results: 6 + 9 = 15.
Answer: 15
For rational functions of the form f(x)/g(x), if direct substitution leads to 0/0, we must factorize the numerator and denominator. By canceling common factors, we remove the discontinuity and evaluate the limit. This technique is vital for handling indeterminate forms.
Step 1: Factorize x² - 1 as (x - 1)(x + 1).
Step 2: Simplify the expression to (x + 1).
Step 3: Substitute x = 1 to get 1 + 1 = 2.
Answer: 2