Math Snap

STEP 1

1. We are given the limit: $\lim _{x \rightarrow 1}\left(\frac{f(x)-f(1)}{x-1}\right)=10$.
2. The function is given as $f(x) = mx^2$.
3. We need to find the value of $m$.

STEP 2

1. Substitute $f(x) = mx^2$ into the limit expression.
2. Calculate $f(1)$.
3. Simplify the limit expression.
4. Solve for $m$ using the given limit value.

STEP 3

Substitute $f(x) = mx^2$ into the limit expression:
$$\lim _{x \rightarrow 1}\left(\frac{mx^2 - f(1)}{x-1}\right) = 10$$

STEP 4

Calculate $f(1)$:
$$f(1) = m(1)^2 = m$$

STEP 5

Substitute $f(1) = m$ into the limit expression:
$$\lim _{x \rightarrow 1}\left(\frac{mx^2 - m}{x-1}\right) = 10$$ Factor out $m$ from the numerator:
$$\lim _{x \rightarrow 1}\left(\frac{m(x^2 - 1)}{x-1}\right) = 10$$ Recognize that $x^2 - 1$ can be factored as $(x-1)(x+1)$:
$$\lim _{x \rightarrow 1}\left(\frac{m(x-1)(x+1)}{x-1}\right) = 10$$ Cancel the $(x-1)$ terms:
$$\lim _{x \rightarrow 1}\left(m(x+1)\right) = 10$$

Evaluate the limit as $x \rightarrow 1$:
$$m(1+1) = 10$$ $$2m = 10$$ Solve for $m$:
$$m = \frac{10}{2}$$ $$m = 5$$ The value of $m$ is:
$$\boxed{5}$$