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Limits and Continuity Test 28

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Limits and Continuity Test 28
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  • Question 1
    1 / -0
    Evaluate the following limits.
    $$\displaystyle\lim_{x\rightarrow a}\dfrac{\sqrt{x}+\sqrt{a}}{x+a}$$.
    Solution
    Evaluate

    $$\displaystyle \lim_{x\to a}\dfrac {\sqrt x +\sqrt a}{x+a}$$

    not an indeterminant Form so, limit can be found by simply putting unit of $$x$$

    $$\displaystyle \lim_{x\to a}\dfrac {\sqrt x +\sqrt a}{x+a}=\dfrac {2\sqrt a}{2a}=\dfrac {1}{\sqrt a}$$
  • Question 2
    1 / -0
    Evaluate the following limits.
    $$\displaystyle\lim_{x\rightarrow 0}\dfrac{3x+1}{x+3}$$.
    Solution
    Evaluate

    $$\displaystyle \lim _{ x\rightarrow 0} \dfrac {3x+1}{x+3}$$

    as $$\to 0, Expression \to \dfrac {1}{3}$$, not indeterminant form

    so, Limit can be foord, by simple substituting value

    $$\displaystyle \lim _{ x\rightarrow 0} \dfrac {3x+1}{x+3}=\dfrac {1}{3}$$
  • Question 3
    1 / -0
    Evaluate the following limits.
    $$\displaystyle\lim_{x\rightarrow 0}\dfrac{x^{2/3}-9}{x-27}$$.
    Solution

  • Question 4
    1 / -0
    Evaluate the following limits.
    $$\displaystyle\lim_{x\rightarrow 1}\dfrac{\sqrt{x^2-1}+\sqrt{x-1}}{\sqrt{x^2-1}}, x > 1$$.
    Solution

  • Question 5
    1 / -0
    Evaluate the following limits.
    $$\displaystyle\lim_{x\rightarrow 0}\dfrac{ax+b}{cx+d}, d\neq 0$$.
    Solution
    $$\displaystyle \lim_{x \to 0}\dfrac {ax+b}{cx+d}, d\neq 0$$

    as $$x\to 0$$, Expression $$\to \dfrac {b}{d}$$, which is a determined Form, so Limit= value

    $$\displaystyle \lim_{x \to 0}\dfrac {ax+b}{cx+d}=\dfrac {b}{a}$$

  • Question 6
    1 / -0
    Evaluate the following limits.
    $$\displaystyle\lim_{x\rightarrow 0}\dfrac{\sqrt{a^2+x^2}-a}{x^2}$$.
    Solution
    $$\displaystyle\lim_{x\rightarrow 0}\dfrac{\sqrt{a^{2}+x^{2}}-a}{x^{2}}$$

    As $$x\rightarrow 0$$, it is $$\dfrac{0}{0}$$ Form, 

    So, using rationalisation

    $$\displaystyle\lim_{x\rightarrow 0}\dfrac{(\sqrt{a^{2}+x^{2}}-a)}{x^{2}}\times \dfrac{(\sqrt{a^{2}+x^{2}}+a)}{\sqrt{a^{2}+x^{2}}+a)}$$

    $$\displaystyle\lim_{x\rightarrow 0}\dfrac{(a^{2}+x^{2})-a^{2}}{x^{2}(\sqrt{a^{2}+x^{2}}+a)}$$             $$((a+b)(a+b)=a^{2}-b^{2})$$

    $$\displaystyle\lim_{x\rightarrow 0}\dfrac{x^{2}}{x^{2}(\sqrt{a^{2}+x^{2}}+a)}$$

    $$\displaystyle\lim_{x\rightarrow 0}\dfrac{1}{(\sqrt{a^{2}+x^{2}}+a)}=\dfrac {1}{\sqrt {a^2}+a}=\dfrac{1}{2a}$$
  • Question 7
    1 / -0
    Evaluate the following limits.
    $$\displaystyle\lim_{x\rightarrow 0}\dfrac{2x}{\sqrt{a+x}-\sqrt{a-x}}$$.
    Solution

  • Question 8
    1 / -0
    Evaluate the following limits.
    $$\displaystyle\lim_{x\rightarrow 0}\dfrac{\sqrt{a+x}-\sqrt{a}}{x\sqrt{a^2+ax}}$$.
    Solution
    $$\displaystyle \lim_{x \rightarrow 0} \dfrac{\sqrt{a+x}-\sqrt{a}}{x \sqrt{a^{2}+ ax}}$$

    as $$x \rightarrow 0$$, it is $$\dfrac{0}{0}$$ form 

    So, using rationalization 

    $$\displaystyle \lim_{x \rightarrow 0} \dfrac{\sqrt{a+x}- \sqrt{a}}{x \sqrt{a^{2}+ax}} \dfrac{(\sqrt{a+x}+ \sqrt{a})}{(\sqrt{a+x}+ \sqrt{a})}$$

    $$\displaystyle \lim_{x \rightarrow 0} \dfrac{x}{x \sqrt{a^{2}+ax} (\sqrt{a+x}+ \sqrt{a})}$$

    $$\displaystyle \lim_{x \rightarrow 0} \dfrac{1}{ \sqrt{a^{2}+ax} (\sqrt{a+x}+ \sqrt{a})}$$

    $$= \dfrac{1}{2a \sqrt{a}}$$
  • Question 9
    1 / -0
    Evaluate the following limits.
    $$\displaystyle\lim_{x\rightarrow 2}\dfrac{x-2}{\sqrt{x}-\sqrt{2}}$$.
    Solution
    $$\displaystyle\lim_{x\rightarrow 2}\dfrac{(x-2)}{\sqrt{x}-\sqrt{2}}$$

    as $$x\rightarrow 2$$, it is $$\dfrac{0}{0}$$ Form, 

    So, using Factorisation

    $$\displaystyle\lim_{x\rightarrow 2}\dfrac{((\sqrt x)^2-(\sqrt 2)^2)}{\sqrt{x}-\sqrt{2}}=\displaystyle\lim_{x\rightarrow 2}\dfrac{(\sqrt{x}-\sqrt{2})(\sqrt{x}+\sqrt{2})}{(\sqrt{x}-\sqrt{2})}$$

    $$\displaystyle \lim_{x\to 2}\sqrt x +\sqrt 2=2\sqrt{2}$$
  • Question 10
    1 / -0
    Evaluate the following limits.
    $$\displaystyle\lim_{x\rightarrow a}\dfrac{x-a}{\sqrt{x}-\sqrt{a}}$$.
    Solution
    $$\displaystyle \lim_{x\rightarrow a}{\dfrac{x-a}{\sqrt x-\sqrt a}}$$

    if $$x\rightarrow a$$, expression $$\rightarrow \dfrac{0}{0}$$, an indetermined form 

    using factorisation

    $$\displaystyle \lim_{x\rightarrow a}{\dfrac{(\sqrt{x}-\sqrt a)(\sqrt x+\sqrt a)}{(\sqrt x-\sqrt a)}}=2\sqrt a$$
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