limx→0√x2+25−5x

The given limit is:

lim (x→0) [√(x²+25) - 5/x]

To solve this limit, we can use the conjugate method. The conjugate of a binomial (a + b) is (a - b). So, the conjugate of [√(x²+25) - 5/x] is [√(x²+25) + 5/x].

Step 1: Multiply the expression by its conjugate over itself:

lim (x→0) [√(x²+25) - 5/x] * [√(x²+25) + 5/x] / [√(x²+25) + 5/x]

Step 2: Simplify the expression:

lim (x→0) [(x²+25 - 25) / x(√(x²+25) + 5/x)]

Step 3: Simplify further:

lim (x→0) [x / (√(x²+25) + 5/x)]

Step 4: As x approaches 0, the expression becomes:

0 / (√25 + 5/0)

Step 5: Simplify the denominator:

0 / (5 + ∞)

Step 6: Any number divided by infinity is 0. So, the limit is 0.