WordGPT

Limit of (1+x)4−1x\frac{\left(1 + x\right)^{4} - 1}{x} as x→0x \to 0

The limit of (1+x)4−1x\frac{\left(1 + x\right)^{4} - 1}{x} as x→0x \to 0 is 44.

Explain it step by step in chatMake a short video of it
Graph of ((1 + x)^4 - 1) / x near x = 0-1.5-1-0.50.511.5510152025limit = 4f(x)
Graph of ((1 + x)^4 - 1) / x near x = 0. The open circle marks the point x = 0, where the function may be undefined; the curve still heads to 4.

How to work it out

  1. 1

    We want what ((1 + x)^4 - 1) / x approaches as x → 0.

    lim⁡x→0(1+x)4−1x\lim_{x \to 0} \frac{\left(1 + x\right)^{4} - 1}{x}
  2. 2

    Direct substitution gives an indeterminate form: 0/0. It does not tell us the limit.

  3. 3

    L'Hôpital's rule: when the form is 0/0, the limit equals the limit of the ratio of the derivatives.

    lim⁡(1+x)4−1x=lim⁡4(1+x)31\lim \frac{\left(1 + x\right)^{4} - 1}{x} = \lim \frac{4 \left(1 + x\right)^{3}}{1}
  4. 4

    Now substitute x = 0: the denominator is not 0 any more.

    41=4\frac{4}{1} = 4
  5. 5

    Method: the derivative of (1+x)^4 at 0 is 4.

  6. 6

    Result:

    lim⁡x→0(1+x)4−1x=4\lim_{x \to 0} \frac{\left(1 + x\right)^{4} - 1}{x} = 4
LaTeX source of the formulas on this page
\[ \lim_{x \to 0} \frac{\left(1 + x\right)^{4} - 1}{x} = 4 \]
\[ \lim_{x \to 0} \frac{\left(1 + x\right)^{4} - 1}{x} \]
\[ \lim \frac{\left(1 + x\right)^{4} - 1}{x} = \lim \frac{4 \left(1 + x\right)^{3}}{1} \]
\[ \frac{4}{1} = 4 \]

Values of ((1 + x)^4 - 1) / x as x → 0 (computed)

xf(x)Distance from the limit
0.14.64100006.41e-1
-0.13.43900005.61e-1
0.014.06040106.04e-2
-0.013.94039905.96e-2
0.0014.00600406.00e-3
-0.0013.99400406.00e-3
0.00014.00060006.00e-4
-0.00013.99940006.00e-4
0.000014.00006006.00e-5
-0.000013.99994006.00e-5

Questions

What is the limit of ((1 + x)^4 - 1) / x as x approaches 0?

The limit of (1+x)4−1x\frac{\left(1 + x\right)^{4} - 1}{x} as x→0x \to 0 is 44.

Which form is it?

0/0.

More well-known limits

    Limit of ((1 + x)^4 - 1) / x as x → 0: 4 | WordGPT