5.0 Algebra of limits

  Limits

• $\mathop {\lim }\limits_{x \to a} \lambda f(x) = \lambda \mathop {\lim }\limits_{x \to a} f(x) = \lambda {l_1}$, where $\lambda $ is constant
• $\mathop {\lim }\limits_{x \to a} \left( {\alpha f(x) \pm \beta g(x)} \right) = \mathop {\lim }\limits_{x \to a} \alpha f(x) \pm \mathop {\lim }\limits_{x \to a} \beta g(x) = \alpha {l_1} \pm \beta {l_2}$, where $\alpha $ and $\beta $ are constants.
• $\mathop {\lim }\limits_{x \to a} f(x).g(x) = \mathop {\lim }\limits_{x \to a} f(x).\mathop {\lim }\limits_{x \to a} g(x) = {l_1}.{l_2}$
• $\mathop {\lim }\limits_{x \to a} \tfrac{{f(x)}}{{g(x)}} = \tfrac{{\mathop {\lim }\limits_{x \to a} f(x)}}{{\mathop {\lim }\limits_{x \to a} g(x)}} = \tfrac{{{l_1}}}{{{l_2}}},{l_2} \ne .$
• $\mathop {\lim }\limits_{x \to a} f(g(x)) = f(\mathop {\lim }\limits_{x \to a} g(x)) = f({l_2})$, if $f(x)$ is continuous at $x$=${l_2}$.

Question 11.

Find $\mathop {\lim }\limits_{x \to 2} ({x^2} + {x^8})$.

Solution:

$\mathop {\lim }\limits_{x \to 2} ({x^2} + {x^8}) = \mathop {\lim }\limits_{x \to 2} {x^2} + \mathop {\lim }\limits_{x \to 2} {x^8}$

$ = {2^2} + {2^8}$

$ = 4 + 256$

$ = 260$

Question 12.

Evaluate $\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{{\tan x}}$

Solution:

$\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{{\tan x}} = \mathop {\lim }\limits_{x \to 0} \frac{{\frac{{\sin x}}{x}}}{{\frac{{\tan x}}{x}}}$ [dividing numerator and denominator by $x$]

$ = \frac{{\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x}}}{{\mathop {\lim }\limits_{x \to 0} \frac{{\cos x}}{x}}}$

$ = \frac{1}{1}$

$ = 1$