Answer: Given limit, $\begin{array}{l} \mathop {\lim }\limits_{x \to 0} {\left( {\cos x + a\sin bx} \right)^{1/x}} \end{array}$ ...
Evaluate the following limit: $\begin{array}{l} \mathop {\lim }\limits_{x \to 0} {\left( {\cos x + \sin x} \right)^{1/x}}\\ \end{array}$
Answer: Given limit, $\begin{array}{l} \mathop {\lim }\limits_{x \to 0} {\left( {\cos x + \sin x} \right)^{1/x}}\\ \end{array}$ f (x) = cos x + sin x – 1 g (x) =...
Evaluate the following limit: $\begin{array}{l} \mathop {\lim }\limits_{x \to 0} {\left( {\cos x} \right)^{1/\sin x}}\\ \end{array}$
Answer: Given limit, $\begin{array}{l} \mathop {\lim }\limits_{x \to 0} {\left( {\cos x} \right)^{1/\sin x}}\\ \end{array}$
Evaluate the following limit: $\begin{array}{l} \mathop {\lim }\limits_{x \to {0^ + }} {\left\{ {1 + {{\tan }^{\sqrt x }}} \right\}^{1/2x}}\\ \end{array}$
Answer: Given limit, $\begin{array}{l} \mathop {\lim }\limits_{x \to {0^ + }} {\left\{ {1 + {{\tan }^{\sqrt x }}} \right\}^{1/2x}}\\ \end{array}$
Evaluate the following limit: $\begin{array}{l} \mathop {\lim }\limits_{x \to \pi } {\left( {1 – \frac{x}{\pi }} \right)^2}\\ \end{array}$
Answer: Given limit, $\begin{array}{l} \mathop {\lim }\limits_{x \to \pi } {\left( {1 - \frac{x}{\pi }} \right)^2}\\ \end{array}$
Evaluate the following limit
Solution: As per the given question,
Evaluate the following limit
Solution: As per the given question,
Evaluate the following limit
Solution: As per the given question,
Evaluate the following limit
Solution: As per the given question,
Evaluate the following limit
Solution: As per the given question,
Let the limit given below
Solution: As per the given question,
Let f(x) be a function defined by given below limit . Then show that lim x-> 0 f(x) doest not exist.
Solution: As per the given question,
Show that the given limit does not exist
Solution: As per the given question,
Solve:
Solution: As per the given question, So, let \[x\text{ }=\text{ }2\text{ }-\text{ }h,\text{ }where\text{ }h\text{ }=\text{ }0\] Substituting the value of \[x,\]we get
Show that the given limit does not exist
Solution: As per the given question,
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}+{{b}^{x}}-2}{x}\]
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{mx}}-1}{{{b}^{nx}}-1},n\ne 0\]
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}+{{a}^{x}}-2}{{{x}^{2}}}\]
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\log (1+x)}{{{3}^{x}}-1}\]
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{5}^{x}}-1}{\sqrt{4+x}-2}\]
Evaluate the following limits: \[\underset{x\to \pi }{\mathop{\lim }}\,\frac{\sqrt{2+\cos x}-1}{{{(\pi -x)}^{2}}}\]
Evaluate the following limits: \[\underset{x\to \pi /4}{\mathop{\lim }}\,\frac{2-\cos e{{c}^{2}}x}{1-\cot x}\]
Evaluate the following limits: \[\underset{x\to \pi /6}{\mathop{\lim }}\,\frac{{{\cot }^{2}}x-3}{\cos ecx-2}\]
Evaluate the following limits: \[\underset{x\to \pi /4}{\mathop{\lim }}\,\frac{\cos e{{c}^{2}}x-2}{\cot x-1}\]
Evaluate the following limits: \[\underset{x\to \pi }{\mathop{\lim }}\,\frac{1+\cos x}{{{\tan }^{2}}x}\]
Evaluate the following limits: \[\underset{x\to a}{\mathop{\lim }}\,\frac{\cos x-\cos a}{x-a}\]
Evaluate the following limits: \[\underset{x\to \pi /3}{\mathop{\lim }}\,\frac{\sqrt{1-\cos 6x}}{\sqrt{2}(\pi /3-x)}\]
Evaluate the following limits: \[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{{{\cos }^{2}}x}{1-\sin x}\]
Evaluate the following limits: \[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{\sin 2x}{\cos x}\]
Evaluate the following limits: \[\underset{x\to \pi /2}{\mathop{\lim }}\,\left( \frac{\pi }{2}-x \right)\tan x\]
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{3\sin x-4{{\sin }^{3}}x}{x}\]
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x\cos x}{3x}\]
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{x}^{2}}}{\sin {{x}^{2}}}\]
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin {{x}^{0}}}{x}\]
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\operatorname{Sin}3x}{5x}\]
Evaluate the following limits: \[\underset{x\to \infty }{\mathop{\lim }}\,\sqrt{x+1}-\sqrt{x}\]
Evaluate the following limits: \[\underset{x\to \infty }{\mathop{\lim }}\,\sqrt{{{x}^{2}}+cx}-x\]
Evaluate the following limits: \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{5{{x}^{3}}-6}{\sqrt{9+4{{x}^{6}}}}\]
Evaluate the following limits: \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{3{{x}^{3}}-4{{x}^{2}}+6x-1}{2{{x}^{3}}+{{x}^{2}}-5x+7}\]
Evaluate the following limits: \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{(3x-1)(4x-2)}{(x+8)(x-1)}\]
Evaluate the following limits: \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{5/7}}-{{a}^{5/7}}}{{{x}^{2/7}}-{{a}^{2/7}}}\]
Evaluate the following limits: \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{2/7}}-{{a}^{2/7}}}{x-a}\]
Evaluate the following limits: \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{(1+x)}^{6}}-1}{{{(1+x)}^{2}}-1}\]
Evaluate the following limits: \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{(x+2)}^{3/2}}-{{(a+2)}^{3/2}}}{x-a}\]
Evaluate the following limits: \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{(x+2)}^{5/2}}-{{(a+2)}^{5/2}}}{x-a}\]
Evaluate the following limits: \[\underset{x\to 2}{\mathop{\lim }}\,\frac{\sqrt{3-x}-1}{2-x}\]
∴ The value of the given limit is \[1/2\]
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+x}-\sqrt{1-x}}{2x}\]
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{{{a}^{2}}+{{x}^{2}}}-a}{{{x}^{2}}}\]
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2x}{\sqrt{a+x}-\sqrt{a-x}}\]
We need to find the limit of the given equation when \[x\text{ }=>\text{ }0\] Now let us substitute the value of x as \[0\], we get an indeterminate form of \[0/0\]. Let us rationalizing the...
Evaluate the following limits: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+x+{{x}^{2}}}-1}{x}\]
Evaluate the following limits: \[\underset{x\to -1/2}{\mathop{\lim }}\,\frac{8{{x}^{3}}+1}{2x+1}\]
Evaluate the following limits: \[\underset{x\to 2}{\mathop{\lim }}\,\frac{{{x}^{3}}-8}{{{x}^{2}}-4}\]
Evaluate the following limits: \[\underset{x\to 3}{\mathop{\lim }}\,\frac{{{x}^{4}}-81}{{{x}^{2}}-9}\]
Evaluate the following limits: \[\underset{x\to 3}{\mathop{\lim }}\,\frac{{{x}^{2}}-4x+3}{{{x}^{2}}-2x-3}\]
∴ The value of the given limit is \[1/2\]