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Limits At Infinity

Learn limits involving trigonometric functions, practice limits at positive & negative infinity and evaluation of more limits at infinity.

Limits at infinity

Limits at  positive infinity\((+\infty)\)

  • Let \(f\)be a function defined in some interval \((a,\infty)\) then

\(\lim\limits_{x\rightarrow\infty}\;f(x)=L\) means that \(f(x)\) can be made arbitrarily close to L by taking  \(x\) sufficiently large.

  • We say \(f(x)\rightarrow L\) as \(x\rightarrow \infty\).
  • \(\infty\) is not a number.

     

For the function whose graph is shown   \(\lim\limits_{x\rightarrow\infty}=f(x)=5\).\(\)

Limits at minus infinity \((-\infty)\)

  • Let \(f\)be a function defined in some interval \((-\infty,a)\) then

         \(\lim\limits_{x\rightarrow-\infty}\;f(x)=L\) means that \(f(x)\) can be made arbitrarily close to L by taking  \(x\) sufficiently large negative.

  • We say \(f(x)\rightarrow L\) as \(x\rightarrow -\infty\).

Illustration Questions

For the function \(f\) whose graph is shown, find the value of \(\lim\limits_{x\rightarrow \infty}\;f(x)\) .

A 2

B 5

C –2

D 1

×

\(\lim\limits_{x\rightarrow \infty}\;f(x)=\) value to which \(f(x)\) approaches as  \(x\) takes very large positive values.

Graph of \(f\)approaches the limit \(y=2\)  as  \(x\) take very large positive values.

 

 

\(\therefore \lim\limits_{x\rightarrow\infty}f(x)=2\)

For the function \(f\) whose graph is shown, find the value of \(\lim\limits_{x\rightarrow \infty}\;f(x)\) .

image
A

2

.

B

5

C

–2

D

1

Option A is Correct

Theorem 

  • If \(r>0\) is a rational number then \(\lim\limits_{x\to\infty} \dfrac{1}{x^r}=0\).
  • \(\lim\limits_{x\to-\infty} \dfrac{1}{x^r}=0\) (if \(x^r\) is defined)

Illustration Questions

Evaluate \(\lim\limits_{x\to\infty} \left(\dfrac{1}{x^3}\right)\)

A \(0\)

B \(-1\)

C \(\infty\)

D \(2\)

×

\(\lim\limits_{x\to\infty} \dfrac{1}{x^r}=0\) for all  \(r>0\)

\(\therefore\lim\limits_{x\to\infty} \dfrac{1}{x^3}=0 \,\,\,\,(r=3)\)

Evaluate \(\lim\limits_{x\to\infty} \left(\dfrac{1}{x^3}\right)\)

A

\(0\)

.

B

\(-1\)

C

\(\infty\)

D

\(2\)

Option A is Correct

Limits at Positive Infinity 

  • To evaluate  \(\lim\limits_{x\to\infty} \dfrac{P(x)}{Q(x)}\)  where  \(P(x)\) and \(Q(x)\) are polynomials in \(x\) , divide  the numerator and denominator  (i.e  \(P(x)\) and \(Q(x)\) by highest power of  \(x \) that occur in \(Q(x)\) ).
  • Then use the property

                                 \(\lim\limits_{x\to\infty} \dfrac{1}{x^r}=0\)      \((r>0)\) . 

           

Illustration Questions

Evaluate     \(\lim\limits_{x\to\infty} \left(\dfrac{2x^2 +7x-6}{3x^2+8x+1}\right)\)

A \(\dfrac{3}{2}\)

B \(\dfrac{2}{3}\)

C \(\dfrac{7}{8}\)

D \(6\)

×

The highest power  in denominator \(Q(x)\)  is \(x^2\) , divide by \(x^2\) in Numerator and  Denominator.

\(=\lim\limits_{x\to\infty} \left[{\dfrac{\left(\dfrac{2x^2 +7x-6}{x^2}\right)}{\left(\dfrac{{3x^2+8x+1}}{x^2}\right)}}\right]\)

\(=\lim\limits_{x\to\infty} \left(\dfrac{2+\dfrac{7}{x}-\dfrac{6}{x^2}}{3+\dfrac{8}{x}+\dfrac{7}{x^2}}\right)\)

\(=\dfrac{\Bigg(\lim\limits_{x\to\infty}2\Bigg) +\Bigg(\lim\limits_{x\to\infty}\dfrac{7}{x}\Bigg) - \Bigg(\lim\limits_{x\to\infty}\dfrac{6}{x^2}\Bigg)}{\Bigg(\lim\limits_{x\to\infty}3\Bigg) +\Bigg(\lim\limits_{x\to\infty}\dfrac{8}{x}\Bigg) + \Bigg(\lim\limits_{x\to\infty}\dfrac{1}{x^2}\Bigg)}\)

\(= \dfrac{2+0-0}{3+0+0} = \dfrac{2}{3}\)

 

Evaluate     \(\lim\limits_{x\to\infty} \left(\dfrac{2x^2 +7x-6}{3x^2+8x+1}\right)\)

A

\(\dfrac{3}{2}\)

.

B

\(\dfrac{2}{3}\)

C

\(\dfrac{7}{8}\)

D

\(6\)

Option B is Correct

Limits at Negative Infinity 

  • To evaluate  \(\lim\limits_{x\to-\infty} \dfrac{P(x)}{Q(x)}\)  where  \(P(x)\) and \(Q(x)\) are polynomials in \(x\) , divide  the numerator and denominator  (i.e  \(P(x)\) and \(Q(x)\) by highest power of  \(x \) that occur in \(Q(x)\) ).
  • Then ,use the property \(\lim\limits_{x\to-\infty} \dfrac{1}{x^r}=0\)      \((r>0)\) . 

Illustration Questions

Evaluate     \(\lim\limits_{x\to-\infty} \left(\dfrac{2x^2 +7x-6}{3x^2+8x+1}\right)\) .  

A \(\dfrac{3}{2}\)

B \(\dfrac{2}{3}\)

C \(\dfrac{7}{8}\)

D \(6\)

×

The highest power  in denominator \(Q(x)\)  is \(x^2\) , divide by \(x^2\) in Numerator and Denominator.

 

\(=\lim\limits_{x\to-\infty} \left[{\dfrac{\left(\dfrac{2x^2 +7x-6}{x^2}\right)}{\left(\dfrac{{3x^2+8x+1}}{x^2}\right)}}\right]\)

 

\(=\lim\limits_{x\to-\infty} \left(\dfrac{2+\dfrac{7}{x}-\dfrac{6}{x^2}}{3+\dfrac{8}{x}+\dfrac{1}{x^2}}\right)\)

\(=\dfrac{\Bigg(\lim\limits_{x\to-\infty}2\Bigg) +\Bigg(\lim\limits_{x\to-\infty}\dfrac{7}{x}\Bigg) - \Bigg(\lim\limits_{x\to-\infty}\dfrac{6}{x^2}\Bigg)}{\Bigg(\lim\limits_{x\to-\infty}3\Bigg) +\Bigg(\lim\limits_{x\to-\infty}\dfrac{8}{x}\Bigg) + \Bigg(\lim\limits_{x\to-\infty}\dfrac{1}{x^2}\Bigg)}\)

\(= \dfrac{2+0-0}{3+0+0} = \dfrac{2}{3}\)

Evaluate     \(\lim\limits_{x\to-\infty} \left(\dfrac{2x^2 +7x-6}{3x^2+8x+1}\right)\) .  

A

\(\dfrac{3}{2}\)

.

B

\(\dfrac{2}{3}\)

C

\(\dfrac{7}{8}\)

D

\(6\)

Option B is Correct

Evaluation of Limits at Infinity 

  • If we want to evaluate any limit of the form \(\lim \limits _{x\to\infty} \dfrac{f(x)}{g(x)}\) where \(f(x),g(x)\) are function containing powers of \(x\) (not necessarily integer) then we look  for the highest power of \(x\) .
  1. If highest power of \(x\) is present in \(f(x)\) but not in \(g(x)\) we say \(\lim \limits _{x\to\infty} \dfrac{f(x)}{g(x)} =\infty\) (divide numerator and denominator  by that power of \(x\))
  2. If highest power of \(x\) is present  in \(g(x)\) and  not in \(f(x)\) we say  \(\lim \limits _{x\to\infty} \dfrac{f(x)}{g(x)} =0\)  (divide numerator and denominator  by that power of \(x\))
  3. If highest  power  of \(x\) is there in both \(f(x)\) and  \(g(x)\) then divide by that power of \(x\) in numerator and denominator . 

Illustration Questions

Evaluate  \(\lim\limits_{x\to\infty} \left(\dfrac{x^{5/2} +5x^2-3x+7}{x^3+8x+3}\right)\)

A \(\dfrac{1}{2}\)

B \(0\)

C \(\dfrac{5}{2}\)

D \(3\)

×

The highest power of \(x \) in the denominator is \(x^3\) , divide by \(x^3\) in numerator and denominator.

\(= \lim \limits _{x\to\infty} \left(\dfrac{\left(\dfrac{x^{5/2}+5x^2-3x+7}{x^3}\right)}{\left(\dfrac{x^{3}+8x+3}{x^3}\right)}\right) \)

\(=\lim \limits _{x\to\infty} \left(\dfrac{{\dfrac{1}{x^{1/2}}+\dfrac{5}{x}-\dfrac{3}{x^2}+\dfrac{7}{x^3}}}{1+\dfrac{8}{x^2}+\dfrac{7}{x^3}}\right) \)

\(= \dfrac{\left(\lim \limits _{x\to\infty} \,\dfrac{1}{x^{1/_2}}\right) + \left(\lim \limits _{x\to\infty} \,\dfrac{5}{x}\right)- \left(\lim \limits _{x\to\infty} \,\dfrac{3}{x^{2}}\right)+ \left(\lim \limits _{x\to\infty} \,\dfrac{7}{x^{3}}\right)}{ \left(\lim \limits _{x\to\infty} \,1\right) +\left(\lim \limits _{x\to\infty} \,\dfrac{8}{x^{2}}\right)+ \left(\lim \limits _{x\to\infty} \,\dfrac{7}{x^{3}}\right)}\)

\(= \dfrac{0+0-0+0}{1+0+0} = \dfrac{0}{1} = 0\)

Evaluate  \(\lim\limits_{x\to\infty} \left(\dfrac{x^{5/2} +5x^2-3x+7}{x^3+8x+3}\right)\)

A

\(\dfrac{1}{2}\)

.

B

\(0\)

C

\(\dfrac{5}{2}\)

D

\(3\)

Option B is Correct

Limits of the Form infinity minus infinity\((\infty-\infty)\) 

  • When limits of the form \((\infty-\infty)\) is given we first use algebra to create  \(\dfrac{\infty}{\infty}\) form and that proceed  by dividing  an appropriate power of \(x\) . 
  • Appropriate power mean the highest power  of \(x\) that is occurring in numerator and denominator combined . 

Illustration Questions

Evaluate   \(\lim\limits_{x\to\infty}\,\left(\sqrt{x^2+x+7}-\sqrt{x^2+2\,x+3}\right)\)

A \(\dfrac{4}{7}\)

B \(\dfrac{-1}{2}\)

C \(\dfrac{1}{2}\)

D \(\dfrac{5}{3}\)

×

\(\lim\limits_{x\to\infty}\,\left(\sqrt{x^2+x+7}-\sqrt{x^2+2\,x+3}\right) = \infty -\infty\)  form (note that both term are very large when \(x\) is large)

Multiply the numerator and denominator by conjugate radical.

\(\lim\limits_{x\to\infty}\,\left(\sqrt{x^2+x+7}-\sqrt{x^2+2\,x+3}\right) \left(\sqrt{x^2+x+7}+\sqrt{x^2+2\,x+3}\right)\)

\(\left(\sqrt{x^2+x+7}+\sqrt{x^2+2\,x+3}\right) \left((u+v)(u-v)=u^2-v^2\right)\)

\(\Rightarrow\lim\limits_{x\to\infty}\dfrac{{x^2+x+7}-{x^2-2\,x-3}}{\sqrt{x^2+x+7}+\sqrt{x^2+2\,x+3}}\)

\(\Rightarrow\lim\limits_{x\to\infty}\,\dfrac{{(-x+4)}}{\sqrt{x^2+x+7}+\sqrt{x^2+2\,x+3}}\)

Divide numerator and denominator by \(x \) :

\(\Rightarrow\lim\limits_{x\to\infty}\,\dfrac{{-1+\dfrac{4}{x}}}{\sqrt{1+\dfrac{1}{x}+\dfrac{7}{x^2}}+\sqrt{1+\dfrac{2}{x}+\dfrac{3}{x^2}}}\)

\(\Rightarrow\dfrac{-1+0}{\sqrt{1+0+0}+\sqrt{1+0+0}} = -\dfrac{1}{2}\)

 

Evaluate   \(\lim\limits_{x\to\infty}\,\left(\sqrt{x^2+x+7}-\sqrt{x^2+2\,x+3}\right)\)

A

\(\dfrac{4}{7}\)

.

B

\(\dfrac{-1}{2}\)

C

\(\dfrac{1}{2}\)

D

\(\dfrac{5}{3}\)

Option B is Correct

Infinite Limits At Infinity 

  • \(\lim\limits_{x\to\infty} \,f(x) = \infty\) means that values of \(f(x)\) becomes large positive  when  \(x \) takes large positive values .
  • \(\lim\limits_{x\to-\infty} \,f(x) = \infty\) mean that values of \(f(x)\) becomes large positive when  \(x \) takes large negative values .
  • \(\lim\limits_{x\to\infty} \,f(x) =- \infty\) mean that values of \(f(x)\) becomes large negative when  \(x \) takes large positive values .

  • \(\lim\limits_{x\to-\infty} \,f(x) =- \infty\)  mean that values of \(f(x)\) becomes large negative when  \(x \) takes large negative values .

Illustration Questions

Find   \(\lim\limits_{x\to\infty} \,(4x^3-x^5)\) .  

A \(\infty\)

B \(-\infty\)

C \(0\)

D \(\dfrac{1}{2}\)

×

\(\lim\limits_{x\to\infty} \,(4x^3-x^5) = \lim\limits_{x\to\infty} \,\overbrace {x^3}^{large +} (4-\overbrace{x^2}^{large - })\)

 

\(=\infty × -\infty = -\infty\)

Find   \(\lim\limits_{x\to\infty} \,(4x^3-x^5)\) .  

A

\(\infty\)

.

B

\(-\infty\)

C

\(0\)

D

\(\dfrac{1}{2}\)

Option B is Correct

Limits of Inverse Trigonometric Function 

  • Most of the limits which involve function having inverse trigonometric function  are evaluated just by the knowledge of the values of these function for various values of \(x \) 
  •  e.g. \(\lim\limits_{x\to0^+} (tan^{-1}(\ell n \,x)) = tan^{-1} (-\infty) = \dfrac{-\pi}{2}\) because we know that  \(\ell n \,x \to- \infty\) as \(x\to 0^+\) .

Illustration Questions

Evaluate \(\lim\limits_{x\to\infty} sin^{-1} \left(\dfrac{2x^2 +5}{4x^2 +7}\right)\) 

A \(\dfrac{\pi}{3}\)

B \(\dfrac{\pi}{6}\)

C \(\dfrac{\pi}{2}\)

D \(\dfrac{-\pi}{4}\)

×

\(\lim\limits_{x\to \infty} \dfrac{P(x)}{Q(x)} = \dfrac{{\text{Coefficient of highest power of x in P(x)}}}{{\text{Coefficient of highest power of x in Q(x)}}}\)

 

If  \({\text{degree }}P(x) = {\text{degree }}Q(x)\) 

In this case \(\lim\limits_{x\to\infty} sin^{-1} \left(\dfrac{2x^2 +4}{4x^2 +7}\right)\)

\(\Rightarrow \lim\limits_{x\to\infty} sin^{-1} \left(\dfrac{2 +\dfrac{4}{x^2}}{4 +\dfrac{7}{x^2}}\right) = sin ^{-1} \dfrac{2}{4} = sin ^{-1} \dfrac{1}{2}\)

\(\Rightarrow\dfrac{\pi}{6}\)

Evaluate \(\lim\limits_{x\to\infty} sin^{-1} \left(\dfrac{2x^2 +5}{4x^2 +7}\right)\) 

A

\(\dfrac{\pi}{3}\)

.

B

\(\dfrac{\pi}{6}\)

C

\(\dfrac{\pi}{2}\)

D

\(\dfrac{-\pi}{4}\)

Option B is Correct

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