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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 151. |
Evaluate the following limits: `lim_(xrarr1/2)((4x^(2)-1)/(2x-1))` |
| Answer» Correct Answer - 2 | |
| 152. |
Evaluate the following limits: `lim_(xrarr1)((x^(3)-x)/(x-1))` |
| Answer» Correct Answer - 3 | |
| 153. |
Evaluate `lim_(xrarr1)(x^(2)-sqrtx)/(sqrtx-1).` |
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Answer» `lim_(xto1)((x^(2)-sqrtx))/((sqrtx-1))=lim_(xto1){((x^(2)-sqrtx))/((sqrtx-1))xx((sqrtx+1))/((sqrtx+1))xx((x^(2)+sqrtx))/((x^(2)+sqrtx))}` `lim_(xto1)((x^(4)-x)(sqrtx-1))/((x-1)(x^(2)+sqrtx))=lim_(xto1)(x(x^3-1)(sqrtx+1))/((x-1)(x^(2)+sqrtx))` `=lim_(xto1)(x(x-1)(x^(2)+x+1)(sqrtx+1))/((x-1)(x^(2)+sqrtx))` `=lim_(xto1)(x(x^(2)+x+1)(sqrtx+1))/((x^(2)+sqrtx))` `=(1xx(1^(2)+1+)(sqrt1+1))/((1^(2)+sqrt1))=((1xx3xx2)/(2))=3["putting"x=1].` |
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| 154. |
Evaluate the following limits: `lim_(xrarr1)(6x^(2)-4x+3)` |
| Answer» Correct Answer - 5 | |
| 155. |
Evaluate the following limits: `lim_(xrarr1)((x^(n)-1)/(x-1))` |
| Answer» Correct Answer - n | |
| 156. |
Evaluate the following limits: `lim_(xrarr(pi)/(4))(1-tanx)/(1-sqrt2sinx)` |
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Answer» Correct Answer - 2 Given limit `=lim_(xto(pi)/(4))(cosx-sinx)/(cosx.{(1)/(sqrt2)-sinx})=lim_(xto(pi)/(4))(sqrt2{(1)/(sqrt2)cosx-(1)/(sqrt2)sinx})/(sqrt2cosx(sin""(pi)/(4)-sinx))` `=lim_(xto(pi)/(4))({sin""(pi)/(4)cosx-cos""(pi)/(4)sinx})/(cosx(sin""(pi)/(4)-sinx))` `=lim_(xto(pi)/(4))(sin((pi)/(4)-x))/(cosx.2cos((pi)/(8)+(x)/(2))sin((pi)/(8)-(x)/(2)))` `=lim_(xto(pi)/(4))(2sin((pi)/(8)-(x)/(2))cos((pi)/(8)-(x)/(2)))/(cosx. 2cos((pi)/(8)+(x)/(2))sin((pi)/(8)-(x)/(2)))=lim_(xto(pi)/(4))(cos((pi)/(8)-(x)/(2)))/(cosx.cos((pi)/(8)+(x)/(2)))` `=(cos((pi)/(8)-(pi)/(8)))/(cos""(pi)/(4),cos""((pi)/(8)+(pi)/(8)))=(cos0)/((cos""(pi)/(4))^(2))=(1)/(((1)/(sqrt2))^(2))=2.` |
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| 157. |
Evaluate the following limits: `lim_(xrarr2)((x^(5)-32)/(x^(3)-8))` |
| Answer» Correct Answer - `20/3` | |
| 158. |
Evaluate the following limits: `lim_(xrarr(pi)/(2))((pi)/(2)-x)tanx` |
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Answer» Correct Answer - 1 Put `((pi)/(2)-x)="y so that when"xto(pi)/(2)then y to0.` `therefore` given limit `=lim_(y to0)y tan((pi)/(2)-y)=lim_(y to0)y cot =lim_(yto0)(y)/(tany)=1.` |
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| 159. |
Evaluate the following limits: `lim_(xrarr0)((sin3x+sin5x))/((sin6x-sin4x))` |
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Answer» Correct Answer - 4 Given limit `lim_(xto0)(2sin4xcosx)/(2cos5x sinx)` `=lim_(xto0){(((sin4x)/(4x)xx4).cosx)/(((sinx)/(x)).cos5x)}=((1xx4)xx1)/(1xx1)=4.` |
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| 160. |
Evaluate the following limits: `lim_(xrarra)((x^(5//2)-a^(5//2))/(x-a))` |
| Answer» Correct Answer - `5/2a^(3//2)` | |
| 161. |
Evaluate the following limits: `lim_(xrarr pi)(sqrt(2+cosx)-1)/((pi-x)^(2))` |
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Answer» Correct Answer - `1/4` Given limit `=lim_(xto(pi)/(4))((sqrt(2+cosx)-1)(sqrt(2+cosx)+1))/((pi-x)^(2)(sqrt(2+cosx)+1))` `=lim_(xto pi)((1+cosx))/((pi-x)^(2)(sqrt(2+cosx)+1))=lim_(hto0)(1+cos(pi-h))/(h^(2){sqrt(2+cos(pi-h))+1})" "["putting"pi -x=h]` `=lim_(hto0)((1-cosh))/{h^(2){sqrt(2-cosh)+1})=lim_(hto0)(2sin^(2)(h//2))/(((h)/(2))^(2)xx4xx{sqrt(2-cosh)+1})` `=1/2lim_(hto0){(sin(h//2))/((h//2))}^(2).(1)/(lim_(hto0){sqrt(2-cosh)+1})=((1)/(2)xx1^(2)xx(1)/(2))=1/4.` |
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| 162. |
Evaluate the following limits: `lim_(xrarr(pi)/(4))((cosec^(2)x-2))/((cotx-1))` |
| Answer» Correct Answer - 2 | |
| 163. |
`lim_(x->pi)((sin3x-3sinx)/((pi-x)^3))` |
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Answer» Correct Answer - `-1` Given limit `=lim_(xto pi)((3sinx-4sin^(3)x-3sinx))/((pi-x)^(3))` `=lim_(xto pi)(-4sinx)/((pi-x)^(3))=lim_(thetato0)(-4sin^(3)(pi+theta))/((-theta)^(3))where (x-pi)=theta` `=-4lim_(thetato0)((sintheta)/(theta))=(-4xx1)=-4.` |
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| 164. |
Evaluate the following limits: `lim_(xrarr(pi)/(2))(sqrt2-sqrt(1+sinx))/(sqrt2cos^(2)x)` |
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Answer» Correct Answer - `1/8` Given limit `=lim_(xto(pi)/(2))((sqrt2-sqrt(1+sinx)))/(sqrt2cos^(2)x)xx((sqrt2+sqrt(1+sinx)))/((sqrt2+sqrt(1+sinx)))` `=lim_(xto(pi)/(2))(2-(1+sinx))/(sqrt2(1-sin^(2)x))xx(1)/((sqrt2+sqrt(1+sinx)))` `lim_(x to(pi)/(2))((1-sinx))/(sqrt2(1-sinx)(1+sinx))xx(1)/((sqrt2+sqrt(1+sinx)))` `=(1)/(sqrt2).lim_(xto(pi)/(2))(1)/((1+sinx)(sqrt2+sqrt(1+sinx)))` `=(1)/(sqrt2)xx(1)/(2xx2sqrt2)=1/8.` |
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| 165. |
Evaluate the following limits: `lim_(xrarra)((sinx-sina))/((x-a))` |
| Answer» Correct Answer - `cosa` | |
| 166. |
Evaluate the following limits: `lim_(x to(pi)/(6))(2sin^(2)x+sinx-1)/(2sin^(2)x-3sinx+1)` |
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Answer» Correct Answer - `-3` Given limit `=lim_(xto(pi)/(6))(2sin^(2)+2sinx-sinx-1)/(2sin^(2)x-2sinx-sinx+1)` `=lim_(xto(pi)/(6))((sinx+1)(2sinx-1))/((sinx-1)(2sinx-1))` `=lim_(xto(pi)/(6))((sinx+1)(2sinx-1))/((sinx-1)(2sinx-1))=lim_(xto(pi)/(6))((sinx+1))/((sinx-1))` |
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| 167. |
verify the statement true or false.If ` lim_( x to a ) [f(x) g(x)]` exists, then both ` lim_( x to a ) f(x) and lim_( x to a ) g (x) ` exist. |
| Answer» If ` underset(x to a) lim [f(x) g(x)]` exists, then both `underset(x to a) lim f(x) and underset( xto a) lim ` g(x) may or may not exist. Hence, it is a false statement. | |
| 168. |
Evaluate the following limits: `lim_(xrarr0)((cosecx-cotx))/(x^(3))` |
| Answer» Correct Answer - 4 | |
| 169. |
Evaluate the following limits: `lim_(xrarr0)((tan2x-sin2x))/(x^(3))` |
| Answer» Correct Answer - 4 | |
| 170. |
Evaluate the following limits: `lim_(xrarr0)((tanx-sinx))/(sin^(3)x)` |
| Answer» Correct Answer - `1/2` | |
| 171. |
Show that `("lim")_(x->0)x/(|x|)`does not exist. |
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Answer» Let `f(x)=(x)/(|x|).` Then, `underset(xto0^(+))limf(x)=underset(hto0)limf(0+h)=underset(hto0)limf(h)=underset(hto0)lim(h)/(|h|)=underset(hto0)limh/h=1.` `underset(xto0^(-))limf(x)underset (xto0)limf(0-h)=underset(hto0)limf(-h)=underset(hto0)lim(-h)/(|-h|)=underset(hto0)lim(-h)/(h)=-1.` `thereforeunderset(xto0^(+))limf(x)neunderset(xto0^(-))limf(x).` Hence, `underset(xto0)limf(x)` does not exist. |
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| 172. |
Evaluate the following limits: `lim_(xrarr0)(sin^(2)mx)/(sin^(2)nx)` |
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Answer» Correct Answer - `(m^(2))/(n^(2))` `lim_(xto0)(sin^(2)mx)/(sin^(2)nx)=lim_(xto0){((sinmx)/(mx)xxmx.(sinmx)/(mx)xxmx)/((sinnx)/(nx)xxnx.(sinnx)/(nx)xxnx)}` `=(m^(2))/(n^(2)).(lim_(xto0)((sinmx)/(mx).(sinmx)/(mx)))/(lim_(xto0)((sin)/(nx).(sinnx)/(nx)))=(m^(2))/(n^(2)).(lim_(xto0)(sinmx)/(mx).lim_(xto0)(sinmx)/(mx))/(lim_(xto0)(sinnx)/(nx).lim_(nto0)(sinnx)/(nx))` `=((m^(2))/(n^(2))xx(1xx1)/(1xx1))=(m^(2))/(n^(2)).` |
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| 173. |
Show that `lim_(xto0^(-)) ((e^(1//x)-1)/(e^(1//x)+1))` does not exist. |
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Answer» Let `f(x)=((e^(1//x)-1)/(e^(1//x)+1)).` Then `underset(xto0^(+))limf(x)=underset(h+0)limf(0+h)=underset(hto0)limf(h)` `=lim_(hto0)((e^(1//h)-1)/(e^(1//h)+1))=underset(hto0)lim((1-(1)/(e^(1//h)))/(1+(1)/(e^(1)//h)))=((1-0)/(1+0))=1.` `underset(xto0^(-))limf(x)underset(hto0)limf(0-h)=underset(hto0)limf(-h)` `=underset(xto0)lim((e^(-1//h)-1)/(e^(-1//h)+1))=underset(hto0)lim(((1)/(e^(1//h))-1)/((1)/(e^(1//h))+1))=((0-1)/(0+1))=-1.` Thus, `underset(xto0^(+))limf(x)neunderset(xto0^(-))limf(x).` Hence, `underset(xto0)limf(x)` does not exist. |
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| 174. |
Let `f(x)={{:(mx^(2)+n",",xlt0),(nx+m",",0lexle1),(nx^(3)+m",",xgt1):}` For what vlaues of intergeras of integers a and n, `lim_(xto0) f(x)and lim_(xto1) f(x)` both exist? |
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Answer» We have `underset(xto0^(+))limf(x)=underset(hto0)limf(0+h)underset(hto0)limf(h)=underset(hto0)lim(nh+m)=m.` `underset(xto0^(-))limf(x)=underset(hto0)limf(0-h)=underset(hto0)limf(-h)=underset(hto0)lim"{"m(-h)^(2)+n"}"=underset(hto0)lim(mh^(2)+n)=n.` ` therefore underset(xto0)limf(x)` exist only when `m=n.` Now, `underset(xto1^(+))limf(x)=underset(hto0)limf(1+h)=underset(xto0)lim"{"n(12+h)^(3)+m"}"=underset(hto0)lim{n(1+h^(3)+3h+3h^(2))+m}=(n+m).` And, `underset(xto1^(-))limf(x)=underset(hto0)limf(1-h)=underset(hto0)lim{n(1-h)+m}=(n+m).` `thereforeunderset(xto1)limf(x)=(n+m).` Hence `underset(xto0)f(x)and underset(xto1)limf(x)` both exist only when `m=n.` |
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| 175. |
If f is an even function, prove that `lim_(xto0^(-)) (x)=lim_(xto0^(+)) f(x).` |
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Answer» Let f be an even function Then, `underset(xto0^(-))limf(x)=underset(hto0)limf(0-h)=underset(hto0)limf(-h)` `=underset(hto0)limf(h)" "[becausef"being even,"f(-h=f(h)]=underset(hto0)limf(0+h)=underset(xto0^(+))limf(x).` Hence, `underset(xto0^(-))limf(x)=underset(xto0^(+))limf(x).` |
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| 176. |
Let = `lim_(xto0)a-(sqrt(a^(2)-x^(2))-(x^(2))/(4))/(x^(4)),agt0.` If is fintine, thenA. a = 2B. a = 1C. `L = (1)/(64)`D. `L=(1)/(32)` |
| Answer» Correct Answer - A,C | |