Questions: (d) lim as x approaches 0 of 3 cos(3x) (e) lim as x approaches 0 of (4x^2) / (x^2 - 3x) (f) lim as t approaches 4 of (t^2 - 16) / (t - 4) (g) lim as t approaches 2 of (5t) / (t - 2) (h) lim as t approaches 4 of 1 / (t(t - 4))

(d) lim as x approaches 0 of 3 cos(3x)
(e) lim as x approaches 0 of (4x^2) / (x^2 - 3x)
(f) lim as t approaches 4 of (t^2 - 16) / (t - 4)
(g) lim as t approaches 2 of (5t) / (t - 2)
(h) lim as t approaches 4 of 1 / (t(t - 4))

Transcript text: (d) $\lim _{x \rightarrow 0} 3 \cos (3 x)$ (e) $\lim _{x \rightarrow 0} \frac{4 x^{2}}{x^{2}-3 x}$ (f) $\lim _{t \rightarrow 4} \frac{t^{2}-16}{t-4}$ (g) $\lim _{t \rightarrow 2} \frac{5 t}{t-2}$ (h) $\lim _{t \rightarrow 4} \frac{1}{t(t-4)}$

Solution

Solution Steps

(d) $\lim _{x \rightarrow 0} 3 \cos (3 x)$

To find the limit as $x$ approaches 0, we can directly substitute $x = 0$ into the function since cosine is continuous.

(e) $\lim _{x \rightarrow 0} \frac{4 x^{2}}{x^{2}-3 x}$

To find this limit, we can simplify the expression and then substitute $x = 0$.

(f) $\lim _{t \rightarrow 4} \frac{t^{2}-16}{t-4}$

This limit can be evaluated by factoring the numerator and then simplifying the expression before substituting $t = 4$.

Step 1: Evaluate $\lim _{x \rightarrow 0} 3 \cos (3 x)$

Since $\cos(x)$ is continuous, we can directly substitute $x = 0$: \[ \lim _{x \rightarrow 0} 3 \cos (3 x) = 3 \cos(0) = 3 \cdot 1 = 3 \]

Step 2: Evaluate $\lim _{x \rightarrow 0} \frac{4 x^{2}}{x^{2}-3 x}$

First, simplify the expression: \[ \frac{4 x^{2}}{x^{2}-3 x} = \frac{4 x^{2}}{x(x-3)} \] As $x \rightarrow 0$, the numerator $4x^2$ approaches 0 faster than the denominator $x(x-3)$, so: \[ \lim _{x \rightarrow 0} \frac{4 x^{2}}{x(x-3)} = 0 \]

Step 3: Evaluate $\lim _{t \rightarrow 4} \frac{t^{2}-16}{t-4}$

Factor the numerator: \[ t^{2} - 16 = (t-4)(t+4) \] Thus, the expression simplifies to: \[ \frac{t^{2}-16}{t-4} = \frac{(t-4)(t+4)}{t-4} = t + 4 \quad \text{for} \quad t \neq 4 \] Now, substitute $t = 4$: \[ \lim _{t \rightarrow 4} (t + 4) = 4 + 4 = 8 \]