Limits Formula for class 11 Maths

Question

Learn Complete Limits Formula for class 11 Maths here, This will so much helpful for students.

Answer 1

Limits:

As we see in everyday life that everything can't be exact. The length of an object can't be a whole number. When we say this rope is of length 5 metre it doesn’t meet with the fact that it is exactly 5 metre. It can be of 4.999999 meters or maybe 5.000001 meters. To deal with such situation limits is introduced in calculus. In this section, we will read limits formula and its properties.

Left and Right-Hand Limits:

1. Left-Hand Limits:

When the expected value of the function is denoted by the points to the left of a fixed point defines the left-hand limit of the function at that point.

2. Right-Hand Limits:

When the expected value of any function is at right side of the fixed point than we say it as right-hand limit of that point.

Representation of limit:

A limit can be normally expressed as:
\({\displaystyle \lim _{n\to c}f(n)=L}\)

Properties and algebra of limits:

Let p and q be two functions and a be a value such that \(\displaystyle{\lim_{x \to a}f(x)}\) and \(\displaystyle{\lim_{x \to a}g(x)}\) exists at that value:

1. Limit of sum of two functions is sum of the limits of the functions, i.e.,

 \(\displaystyle{\lim_{x \to a}[f(x) + g (x)] = \lim_{x \to a}f(x) + \lim_{x \to a}g (x)}\)

2. Limits formula of difference of two functions is difference of the limits of the functions, i.e.,

 \(\displaystyle{\lim_{x \to a}[f(x) – g (x)] = \lim_{x \to a}f(x) – \lim_{x \to a}g (x)}\)

3. Limits for any real number k, \(\displaystyle{\lim_{x \to a}[k f(x)] = k \lim_{x \to a}f(x)}\)
4. Limits formula of product of two functions is product of the limits of the functions, i.e.,

 \(\displaystyle{\lim_{x \to a}[f(x)\; g(x)] = \lim_{x \to a}f(x) \times \lim_{x \to a}g(x)}\)

5. Limit of quotient of two functions is quotient of the limits of the functions (whenever the denominator is non zero), i.e.,

 \(\displaystyle{\lim_{x \to a}\frac{f(x)}{g(x)}} =\frac{\displaystyle{\lim_{x \to a}f(x)}}{\displaystyle{\lim_{x \to a}g(x)}}\)

Standard Limits:

1. \(\displaystyle{\lim_{x \to a}\frac{x^{n} – a^{n}}{x - a}}\) = na^n-1
2. \(\displaystyle{\lim_{x \to 0}\frac{sin x}{x}}\) = 1
3. Also, \(\displaystyle{\lim_{x \to 0}\frac{1 – cos x}{x}}\) = 0

Limits formula based examples:

1. Find the limit \(\displaystyle{\lim_{x \to 1}\:[x^{3} – x^{2} + 1]}\).

Solution:
\(\displaystyle{\lim_{x \to 1}}\) [x³ – x² + 1] = 1³ – 1² + 1 = 1

2. Find the limit \(\displaystyle{\lim_{x \to 1}\frac{x^{2} + 1}{x + 100}}\).

Solution:
\(\displaystyle{\lim_{x \to 1}\frac{x^{2} + 1}{x + 100}}\) = \(\frac{1^{2} + 1}{1 + 100}\) =  \(\frac{2}{101}\)