NCERT Solutions
Class 11 - Mathematics - Chapter 12: LIMITS AND DERIVATIVES

EXERCISE 12.1

Evaluate \(\lim_{x\to 3} (x+3)\).

Evaluate \(\lim_{x\to \pi} (x - \tfrac{22}{7})\).

Evaluate \(\lim_{r\to 1} \pi r^{2}\).

Evaluate \(\lim_{x\to 4} \dfrac{4x+3}{x-2}\).

Evaluate \(\lim_{x\to 1} \dfrac{x^{10}+x^{5}+1}{x-1}\).

Evaluate \(\lim_{x\to 0} \dfrac{(x+1)^{5}-1}{x}\).

Evaluate \(\lim_{x\to 2} \dfrac{3x^{2}-x-10}{x^{2}-4}\).

Evaluate \(\lim_{x\to 3} \dfrac{x^{4}-81}{2x^{2}-5x-3}\).

Evaluate \(\lim_{x\to 0} \dfrac{ax+b}{cx+1}\).

Evaluate \(\lim_{z\to 1} \dfrac{z^{1/3}-1}{z^{1/6}-1}\).

Evaluate \(\lim_{x\to 1} \dfrac{ax^{2}+bx+c}{cx^{2}+bx+a}\), where \(a+b+c\neq 0\).

Evaluate \(\lim_{x\to -2} \dfrac{\tfrac{1}{x}+\tfrac{1}{2}}{x+2}\).

Evaluate \(\lim_{x\to 0} \dfrac{\sin ax}{bx}\).

Evaluate \(\lim_{x\to 0} \dfrac{\sin ax}{\sin bx}\), where \(a,b\neq 0\).

Evaluate \(\lim_{x\to \pi} \dfrac{\sin(\pi - x)}{\pi(\pi - x)}\).

Evaluate \(\lim_{x\to 0} \dfrac{\cos x}{\pi - x}\).

Evaluate \(\lim_{x\to 0} \dfrac{\cos 2x - 1}{\cos x - 1}\).

Evaluate \(\lim_{x\to 0} \dfrac{ax + x\cos x}{b\sin x}\).

Evaluate \(\lim_{x\to 0} x\sec x\).

Evaluate \(\lim_{x\to 0} \dfrac{\sin ax + bx}{ax + \sin bx}\), where \(a, b, a+b \neq 0\).

Evaluate \(\lim_{x\to 0} (\csc x - \cot x)\).

Evaluate \(\lim_{x\to \pi/2} \dfrac{\tan 2x}{x - \pi/2}\).

Find \(\lim_{x\to 0} f(x)\) and \(\lim_{x\to 1} f(x)\), where

\(f(x) = \begin{cases} 2x+3, & x \le 0 \\ 3(x+1), & x>0 \end{cases}\).

Find \(\lim_{x\to 1} f(x)\), where

\(f(x) = \begin{cases} x^{2}-1, & x \le 1 \\ -x^{2}-1, & x>1 \end{cases}\).

Evaluate \(\lim_{x\to 0} f(x)\), where

\(f(x) = \begin{cases} \dfrac{|x|}{x}, & x \ne 0 \\ 0, & x=0 \end{cases}\).

Find \(\lim_{x\to 0} f(x)\), where

\(f(x) = \begin{cases} \dfrac{x}{|x|}, & x \ne 0 \\ 0, & x=0 \end{cases}\).

Find \(\lim_{x\to 5} f(x)\), where \(f(x) = |x| - 5\).

Suppose

\(f(x) = \begin{cases} a+bx, & x<1 \\ 4, & x=1 \\ b-ax, & x>1 \end{cases}\)

and if \(\lim_{x\to 1} f(x) = f(1)\), what are possible values of \(a\) and \(b\)?

Let \(a_{1}, a_{2}, \ldots, a_{n}\) be fixed real numbers and define

\(f(x) = (x-a_{1})(x-a_{2})\cdots(x-a_{n})\).

What is \(\lim_{x\to a_{1}} f(x)\)? For some \(a \ne a_{1}, a_{2}, \ldots, a_{n}\), compute \(\lim_{x\to a} f(x)\).

If

\(f(x) = \begin{cases} |x|+1, & x<0 \\ 0, & x=0 \\ |x|-1, & x>0 \end{cases}\),

for what value(s) of \(a\) does \(\lim_{x\to a} f(x)\) exist?

If the function \(f(x)\) satisfies

\(\displaystyle \lim_{x\to 1} \dfrac{f(x) - 2}{x^{2} - 1} = \pi\),

evaluate \(\lim_{x\to 1} f(x)\).

If

\(f(x) = \begin{cases} mx^{2}+n, & x<0 \\ nx+m, & 0 \le x \le 1 \\ nx^{3}+m, & x>1 \end{cases}\).

For what integers \(m\) and \(n\) does both \(\lim_{x\to 0} f(x)\) and \(\lim_{x\to 1} f(x)\) exist?