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Find the value of following limits : Limit x tends 0 ( 7^x -3^x)/(5^x -2^x)

Find the value of following limits : Limit x tends 0 ( 7^x -3^x)/(5^x -2^x)

Question Image
Grade:11

2 Answers

Kailash Jagadeesh
35 Points
3 years ago
Take 3^x common in the numerator and 2^x in the denominator and write the numerator and denominator as separate fractions . Now multiply and divide by x and use the identity: Lim x---> 0 a^x-1/x = ln a
Vikas TU
14149 Points
3 years ago
Dear Student,
limx->0 (7x – 3x)/(5x – 2x) = limx->0 (7xln7 – 3xln3)/(5xln5 – 2xln2) = (ln7 – ln3)/(ln5 – ln2)
= ln(7/3)/ln(5/2) = ln(7/3) – ln(5/2).
134.
y'' = 2e-x ------ (1)
y(0) = 1 , y'(0) = 0.
Integrating (1) wrt x, we get y' = -2e-x + c.
Applying the IC y'(0) = 0, we get y'(0) = -2 + c = 0. Therefore c = 2.
Integrating y' = -2e-x + 2 wrt x, we get y = 2e-x + 2x + d.
Applying the IC y(0) = 1, we get y(0) = 2 + d =  1. Therefore d = -1.
Therefore the solution is y = 2e-x + 2x – 1.
Cheers!!
Regards,
Vikas (B. Tech. 4th year
Thapar University)

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