Limit of (x^n-a^n)/(x-a) as x approaches a: Formula, Proof

The limit of (x^n-a^n)/(x-a) as x approaches a is equal to na^n-1. This limit is denoted by lim_x→a (xⁿ-aⁿ)/(x-a), so the limit formula of (xⁿ-aⁿ)/(x-a) when x tends to a is given as follows.

limx→a (xn-an)/(x-a) = n⋅an-1

Lets prove this limit formula.

Proof of lim_x→a (xⁿ-aⁿ)/(x-a)

To prove lim_x→a (xⁿ-aⁿ)/(x-a) = n⋅a^n-1 we will consider three different cases depending upon n is a positive integer, negative integer, or a rational number.

Case 1:

n is a positive integer.

Using the binomial identity: xⁿ-aⁿ = (x-a) (x^n-1 +x^n-2a +…+a^n-1) we get that

lim_x→a (xⁿ-aⁿ)/(x-a)

= lim_x→a (x^n-1 +x^n-2a +…+a^n-1)

= a^n-1 +a^n-2a +…+a^n-1

= a^n-1 +a^n-1 +…+a^n-1 {n times}

= na^n-1.

So the limit of (xⁿ-aⁿ)/(x-a) is equal to na^n-1 when x tends to a.

Case 2:

n is a negative integer.

Assume n= -m where m is a positive integer.

So the limit formula lim_x→a (xⁿ-aⁿ)/(x-a) = na^n-1 is valid when n is a negative integer.

Case 3:

n is any real number.

Assume n= p/q where q≠1 is a positive integer and p is either a positive or a negative integer.

Let x^1/q = z and a^1/q =b.

Therefore, x=z^q and a=b^q.

Observe, z→b when x→a.

Now,

$\dfrac{x^n-a^n}{x-a}$ $=\dfrac{x^{p/q}-a^{p/q}}{x-a}$ $=\dfrac{z^p-b^p}{z^q-b^q}$ $=\dfrac{\frac{z^p-b^p}{z-b}} {\frac{z^q-b^q}{z-b} }$

Therefore,

lim_x→a (xⁿ-aⁿ)/(x-a)

= lim_z→b [(z^p-b^p)/(z-b) / (z^q-b^q)/(z-b)]

= lim_z→b (z^p-b^p)/(z-b) / lim_z→b (z^q-b^q)/(z-b)

= pb^p-1 /qb^q-1

= $\dfrac{p}{q}b^{p-q}$

= $\dfrac{p}{q}a^{\frac{p-q}{q}}$ since a^1/q =b.

= $\dfrac{p}{q}a^{\frac{p}{q}-1}$

= na^n-1

So the limit of (xⁿ-aⁿ)/(x-a) is equal to na^n-1 as x approaches a, and this is proved for any real number n. This completes the proof.

More Limits:

Limit of (xⁿ-1)/(x-1) when x→1

Limit of x^1/x when x→∞

Limit of xsin(1/x) when x→0

Limit of x²sin(1/x) when x→0

Limit of 1/x² when x→∞

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