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} (x^{n}-a^{n})/(x-a), so the limit formula of (x^{n}-a^{n})/(x-a) when x tends to a is given as follows.

lim_{x→a} (x^{n}-a^{n})/(x-a) = n⋅a^{n-1} |

Lets prove this limit formula.

## Proof of lim_{x→a} (x^{n}-a^{n})/(x-a)

To prove lim_{x→a} (x^{n}-a^{n})/(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: x^{n}-a^{n} = (x-a) (x^{n-1} +x^{n-2}a +…+a^{n-1}) we get that

lim_{x→a} (x^{n}-a^{n})/(x-a)

= lim_{x→a} (x^{n-1} +x^{n-2}a +…+a^{n-1})

= a^{n-1} +a^{n-2}a +…+a^{n-1}

= a^{n-1} +a^{n-1} +…+a^{n-1} {n times}

= na^{n-1}.

So the limit of (x^{n}-a^{n})/(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} (x^{n}-a^{n})/(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} (x^{n}-a^{n})/(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 (x^{n}-a^{n})/(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 (x^{n}-1)/(x-1) when x→1

Limit of x^{2}sin(1/x) when x→0

## FAQs

### Q1: What is the limit of (x^{n}-a^{n})/(x-a) when x tends to a?

Answer: The limit of (x^{n}-a^{n})/(x-a) is equal to na^{n-1} when x tends to a, that is, lim_{x→a} (x^{n}-a^{n})/(x-a) = na^{n-1}.